The word "democracy" comes from two Greek words:
demo- a combining form occurring in loanwords from Greek, where it meant “people.”
-cracy: a combining form occurring in loanwords from Greek, with the meaning “rule,” “government,” “governing body.”
So democracy literally means: "Rule by the people." Or more specifically, the majority of the people.
In my mind, anyone that aims for us to be ruled, even by "the people" (as abstract and meaningless as that concept is) is not promoting anarchy. But unfortunately this view is not always shared by the people calling themselves anarchists today. It's difficult for me to imagine that an anarchist; who is presumably opposed to authority in all its guises and hopefully rejects the very notion of rulers, would then consent to being ruled by "the people"... I know I sure as hell don't want to be ruled by anyone. But a lot of anarchists continue to romanticize democracy, perhaps because they're unable to break through the years of propaganda fed to them by the state and its schooling and media apparatuses.
From an early age, it's hammered into us that democracy = freedom. Any anarchist will tell you that although most of us live in societies that are governed by forms of democracy, none of us have anything resembling freedom. Yet a lot of us make excuses to ourselves so we can continue to romanticize democracy... Tell a room full of anarchists that you oppose democracy and you'll no doubt hear impassioned insistences that what we have now isn't "real" democracy, but "if we had anarchism, we'd have "real" democracy and things would be different, because anarchism is the only real democracy!"
A lot of anarchists spend a lot of effort holding onto oppressive phantoms like democracy and go through great lengths to fuze these liberal concepts with anarchy, when we really have no reason to. Anarchists who insist anarchy and democracy are one and the same when democracy is responsible for an endless list of horrible atrocities do no service to anarchy.
Our rulers use democracy to separate us into in-groups and out-groups, pitting the majority group against the minority groups and giving everyone a false sense of control. We're made to believe we have a say in how our lives are run because we get to participate in glorious democracy. Of course, all of us outside the ruling class continue to be exploited, living in perpetual servitude, and the only people who really benefit from democracy are the ruling class who use it to keep us alienated and distracted so we don't rise up and kill them all for the debilitating misery they create. Anarchy rejects authority and it rejects the domination of majority groups over minority groups. Anarchy is about upholding each individual's autonomy and dismantling the authority forced on us by oppressive actors.
Democracy grants authority to favored groups to oppress minority groups. Democracy ignores the autonomy of the individual in favor of the collective will of the dominant group. Democracy exists to enable rulers to uphold brutal power hierarchies. It's really the full embodiment of authority; used to maintain the tyrannical capitalist-statist status quo all over the world today.
The Failure of Democracy
Democracy is the tyranny of the majority, however you try to window-dress it. In practice, all forms of democracy have been used by a majority group to control or otherwise dictate to a minority group. All forms of democracy have been used to smother autonomy, to stifle self-determination, and to absolve rulers of responsibility for their actions. How can a ruler be responsible for their atrocities when "the people" elected them and empowered them to commit those atrocities?
Though you'll never hear democracy-fetishists mention it, Hitler was technically democratically elected in accordance with the German political system. His actions after being elected were largely supported by the majority group in Germany. All the atrocities he committed were done on behalf of that majority group; to strengthen the position of "Aryan" Christians in society at the expense of everyone else. The German people empowered Hitler to maintain their privilege at all cost.
There's no reason so-called "real" democracy would be any different than the democracy that created nazi Germany. Participatory democracy would just allow more members of the dominant group to more directly participate in enacting brutal policies.
"Real" democracy won't stop people from choosing to oppress others to benefit their own group. If the majority of WW2 Germans stood by and cheered while people were carted off to concentration camps, why would anyone think "real" democracy would have changed that? Throughout history, whenever a skilled propagandist points the finger at a minority group, the majority group tears them limb from limb. This is democracy in action. White supremacy and even genocide have been propped up with the power of democracy countless times.
Democracy or Anarchy?
So do anarchists support democracy? Not if those anarchists have a fully developed understanding of what anarchy entails. Not if they're serious about liberating themselves from authority and crushing hierarchies as they form.
Democracy is really not compatible with anarchy in any permeable way. It could be a useful process for gauging the views of each member of a small group, but that shouldn't be enough for us to make the claim that "anarchy is democratic". Anarchy is the opposition to authority. It's the struggle against oppression. The quest to limit suffering. We shouldn't be claiming anarchy is defined by democracy; which is a specific system of government that demands people be ruled by other people.
If you ask 10 random anarchists whether they support democracy, you're certain to get a mixed response. Every person you ask will be at a different point on their political journey, and some anarchists will spend a lot more time thinking about labor rights, housing, migrant aid and other pressing concerns, while putting very little philosophical thought into the nature of hierarchy and all the ways it manifests itself and becomes ingrained in our lives.
Collectivist-minded anarchists will usually insist on direct-democracy and consensus-democracy as decision-making mechanisms, but it frequently leads to problems when certain members of the group don't fall in line with the majority's agenda. The bigger the group, the more likely this is to happen. The minority members will inevitably grow frustrated at this oppression and either leave the group or be forced to conform in order to stay.
In practical terms, for example, this could mean all black people in a community could be alienated, marginalized or even forced to leave their homes altogether because the white majority have voted to ignore their concerns in order to safeguard white privilege. Democracy and marginalization tend to come as a group deal. "Power to the people" really means "power to the most powerful group of people", and the more power the powerful group has, the less power the marginalized groups have.
The Authority of Democracy
Western democracy originated in ancient Greece. This political system granted democratic citizenship to free men, while excluding slaves, foreigners and women from political participation. In virtually all democratic governments throughout ancient and modern history, this was what democracy meant. An elite class of free men made all the decisions for everyone. Before Athens adopted democracy, aristocrats ruled society, so "rule by the people", or the idea of a government controlled (in theory) by all its (free) male citizens instead of a few wealthy families seemed like a good deal. But really it was just a new iteration of Aristocracy rule rather than the revolution it's painted as. The rich still rule society by feeding voters carefully constructed propaganda and keeping everyone poor, overworked and desperate to be granted basic needs by the state.
In democracies today, only legal citizens of a country are granted democracy. In a lot of countries, people who have been convicted of a "crime" are denied the right to vote, regardless of how long ago they served their sentence. In the US, this is used to deny voting rights to minority groups, who make up a large proportion of the prison population.
In some societies only a small minority group are allowed to participate in the democracy. In Apartheid South Africa, the minority group (European settlers) granted themselves democracy and excluded the native majority, using democracy to deprive the native population of the rights granted to European settlers. Anarchy, of course, is an absence of government; of rulers. Democracy aims for the individual to be governed, ruled, controlled by others. So its plain to see that anarchy is fundamentally incompatible with democracy.
There has been a lot written outlining why the concept of democracy simply cannot be made compatible with anarchy, yet a lot of people identifying as anarchists today refuse to let go of the idea of democracy as a revolutionary method, and insist it can somehow overcome its inherently hierarchical nature and long history of oppression. In all honestly, a lot of these people are simply confused minarchists that don't actually want to abolish hierarchy; but instead minimize it.
Consensus democracy aims to get everyone in a group to agree to take a unified path of action. It sounds good in theory, but the only way to get everyone from disparate backgrounds and experiences to agree to the same thing is to water down the plan to such an extent that the action will likely become meaningless. Consensus democracy assumes the majority group won't bully or peer pressure the minority group into folding to their will. It ignores the basic reality that some people will aggressively force their will on others, or at least shame or manipulate opponents into submission.
The whole concept of consensus democracy reminds me of that meme with the smug guy sitting at the booth with the "change my mind" sign; inviting his political opponents to debate him. I can safely say if I saw that guy sitting at that booth, I'd walk the other way. Why should anyone be put in a situation where they're forced to expend all their energy to change someone's mind? Just do your own thing and don't worry about people that don't want to participate in what you're doing. If people have fundamental disagreements, then they don't need to cooperate. It's not the end of the world.
Fruitless attempts to get everyone to reach the same agreement is just the latest form of the bureaucratic meandering that has long sabotaged political action. After countless hours of heated debate, and a long series of compromises, the consensus reached (if it's ever reached at all) will likely be very watered down from its initial form and be of little benefit to anyone in the group. A plan for concrete action will have been turned into a frustrating exercise in concession, tepid half-measures, and ultimately; inaction. All because the people who made the plan felt the need to gain the approval of a committee of naysayers before pursuing it.
Anarchy Doesn't Need Democracy
Instead of a large group laboring to make democracy work so they can agree on a course of action, it would be far more productive for smaller groups made up of people with shared interests to splinter off and co-operate to follow their own plans that require no compromise because their interests are already aligned. Throughout history, democracy has existed to legitimize authority, providing justification for hierarchical power structures by framing every oppressive action the state takes against us as "the will of the people". It has long enabled the powerful to crush the powerless. People who insist on associating anarchism with democracy are trying to legitimize anarchism, to associate it with comfortable institutions embraced by thoroughly indoctrinated liberals. But anarchy has no want or need for legitimization. Anarchy doesn't need to be watered down to broaden its appeal to a public that is high on hierarchy.
Anarchists always oppose monarchy; the rule of one. We always oppose oligarchy; the rule of a few. So why wouldn't we oppose democracy; the rule of many? Why should the many get to decide how you or I live our lives? A ruler is a ruler is a ruler. Democracy has been expertly wielded as a weapon by the elites in society. By combining democracy with meticulously-crafted propaganda, the powerful are able to control voters and manipulate them into voting against their own interests.
Democracy has forever been synonymous with class based societies. It has split entire countries into two barely-distinctive political parties (conservative and "progressive") that are nevertheless permanently at each other's throats. Even in its most libertarian-friendly forms, it has constantly failed to avert hierarchy, coercion and the authoritarian machinations of majority-groups.
You can't strive to replace an artificial system as brutally hierarchical as democracy with a supposedly more egalitarian version of the same thing and call it anarchy. You have to throw the whole rotten system out.
Reject democracy. Reject the notion that you should be ruled by anyone. Embrace self-determination. Embrace anarchy.