The big objection is to making majority (or even supermajority) decisions binding on the minority, which is seen as coercive. Smaller objections to things like elected positions and parliamentary procedure, which are seen as too hierarchical.
It seems to me they have a problem with anything, frankly.
Mainly I just wanted to point out that democracy is also a flat system and has no kings.
Still, majority in democracy happens ad hoc. You don't know on which side of the fence you will sit. At least, in the mean, you will agree with the majority.
(And by the way, there is a common folklore that representative democracy can protect minorities better. There are several reasons why this is untrue, and in fact, direct democracy protects minorities better than representation.)
Having no kings is not the same as being flat. Democracies have no kings, because any leaders are elected (instead of inheriting/being installed/etc.). But democracies can be anything but flat. Consider Germany, where you have three tiers of government (municipal, state, and federal), and each tier is further subdivided in its hierarchy of power (e.g. Member of Parliament, cabinet minister, chancellor).
That doesn't mean there cannot be "hierarchy" in the sense of administrative division (state, county) or in the sense of respect (like in Linux kernel development for instance).
In the Athenian democracy minorities were inter alia excluded from political decision making - one had to be a male member of the ethnic majority to participate, there was no mechanism for obtaining citizenship other than blood.
However - in comparison to how other Greek cities treated them, metics (resident aliens) in Athens enjoyed far greater security and had codified rights.
To me this latter point is decisive in showing that the horizon of direct democracy is always towards greater inclusiveness. But one has to make up one's own mind on these matters.
One reason is that representation by definition focuses on people and trust in them (you are selecting "your man"), as opposed to issues. Some politicians try to incite inter-group hatred, because it is a strategy that pays off in this game. This brings out human ingroup/outgroup biases and that then causes issues for the minorities. (We have some evidence that people in Switzerland do not see "losing" in particular referendum as a big deal, they understand that the minority is adhoc.)
The other reason is that representatives are typically given authority over bunch of issues bundled together. So majority might vote in a representative because they agree with their economic policy, but they don't neccessarily agree with his, say, immigration policy. In fact, for less important issues, the elected representative can have completely minority opinions!
That means that unless the majority has horrible opinions (which it rarely has in actuality, and then it's questionable how any form of government arising from that culture could deal with this case anyway), there is always a risk of voting in a representative with a horrible opinion (and authority in that matter). While in direct voting, this issue would have to break the majority boundary, which it often never will. (Direct democracies are quite conservative in the real world.)
An example is Donald Trump's treatment of problem with ICE. Even if large majority of Americans disagree with how ICE treats immigrants, lot of people approved Trump (for other reasons), and he pushed (because what a person he is) for bad policies in that matter.
Furthermore, if you actually use utilitarian morality (which is disputable), and calculate the mean value of the bad policy effect on the minority over all possible distributions of minority and agreement with a horrible policy towards said minority, then you will even get the result that the representation is objectively worse (in the probabilistic sense) than direct vote on the policy. It is because "tyranny of the minority" (selecting a wrong person as a representative) is, it turns out, overall a bigger issue than "tyranny of the majority" (the former can work against both minority and majority, the latter only against a minority).
As I said, I can't give references, but the above would certainly be a worth pursuit. There are some people in direct democracies (Switzerland) studying it, but it is rather small number of people.
