Remember kids, Tankies wants to undermine democracy - same as facists.
The state is needed to enforce private property, not the lack of it.
Critically, you assume that authority is required for order, so obviously you’re unable to envision whatever it might be that anarchists are getting at.
Sure, communism is an ideal that does not exist in practice currently, but the point of an ideal is to work towards it. The “anarcho” part specifies complete opposition to authority in praxis as well. Strategies could include unionising, community building, mutual aid, permablitzing, FOSS, copyleft, and whatever else can undermine the current power structures while maintaining anarchist principles. Which explicitly excludes Marxist-Leninist strategies of coopting the state, or forming any other kind of heirarchy.
Using the state to make private property “illegal” might be a possible strategy, but it certainly isn’t an anarchist one.
Hypothetically, the moment that the state doesn’t exist, private property would be unenforceable/meaningless. How the state might be abolished is another question. While marxist-leninists might opt for a “trust me bro the state will dissolve once we reach true communism” strategy, anarchists again would not. Obviously it’s a tough question and an even tougher endeavour but it’s not something anarchists don’t have answers for.
Anarchist praxis is more grassroots focused, because obviously anarchy/anarcho-communism can’t happen overnight, and requires a profound change in social values. If you choose to believe that it’s impossible, I can’t blame you, but I would like to ask why.
Also you use private property and “property” interchangeably, but there’s a difference between personal property and private property. Abolishing private property isn’t about taking away your personal house and toothbrush, it’s about abolishing landlords, factory owners, and other “private” ownership of public/communal spaces.
Regarding unions, it depends whether or not they’ve been structured specifically without hierarchy in mind. In any case the point is that anarchist praxis is diverse and decentralised.
Actually an entire state made out of democratic worker unions sounds pretty good. Although I personally would prefer no state.