we are simply using a different definition of public and private. i am using the legal one: public is government, private is everything else. i explained why the distinction matters and what effects it has.
your points are valid but arguing about the definition of what is public isn't helpful.
same goes for government control. you are not wrong, but again, the legal difference in what that control is matters. private entities have constitutional protection and rules that are clearly defined and distinct from the protection and rules for public institutions. the control the government has is simply different and that difference matters.
the bigger reach of municipalities doesn't go away in HOAs because most HOAs are within a municipality. that makes HOAs at best redundant, at worst you have to suffer both. at least with municipality you have democracy on your side. HOAs are controlled by owners, everyone else is second class. that's not a democracy. the result is that the combination of government and HOA gives you less freedom than government alone. HOAs don't replace the government, they don't make it go away.
HOAs in my opinion should be treated like government entities because they are given quasi government like powers without the same checks and balances that government has. they aren't and for me that's a problem.