everyone who hates anarchists is a tacit participant in antidisestablishmentarianism
Well, not quite; establishmentarians support the established church, that is, the existence of a church that is a part of the government (such as the Church of England, whose head is the reigning monarch, and whose bishops sit in the legislature, in this case, the House of Lords).
Disestablishmentarianism is support for the removal of this linkage between chuch and state; The establishment clause of the US First Amendment is an excellent example of disestablishmentarian legislation.
So an antidisestablishmentarianist opposes the separation of church and state. While such an individual might consider such separation to be anarchistic, and anarchism to be bad, that's not necessarily implied.
This is typical of English, which due to its mongrel history often has a number of different potential words with identical meanings, and which leverages that redundancy to introduce nuance - "The establishment" can refer to any system that is currently in place - as you observe, from 'esta', to be - but it's derivation 'establishmentarian' has been used specifically to refer to the system of legal synergy between church and state, to the point where (at least in the UK and most commonweath nations), it would normally be taken to refer specifically to that particular established situation.
Indeed, it is such nuances of meaning that leads me to reject bio- or omni- as roots when selecting the word holocide to carry the specific meaning of 'killing all people'.
Both of the other roots would, in my opinion, suggest killing non-human life as well, which isn't what happened in the flood story (another fictional example would be Drax's plot in the movie Moonraker*, where he plans holocide using a gas that specifically kills only humans, so as to preserve the biosphere of the Earth for his select group of 'superior specimens. It's basically the same as the Noah story, but with the moral inverted to portray the guy killing everyone as a villain, rather than a hero).
*The plot of the Ian Fleming novel of the same name is radically different from that of the movie, and in the book the villainous Drax is attempting to nuke London in revenge for their victory over Naziism.