Sorry Max but that argument can be used against any government system.
The problem is that wouldn't the established regimes in most countries see an anarchist "nation" (for lack of a better word) as a threat and thus invade? Also many inside the territory would oppose its system, probably much more than in normal countries. And in the case of companies owned by the workers... a state is still needed to oversee the companies or at least maintain a police and military force. Such things are better off in the hands of a centralized state rather than local militia.
Anyways I'm so tired of talking about politics. I need to take a break from this stuff before I get... heated as usual.
You can say the exact same thing of democracy in general however. Didn't all the other nations in Europe band together to invade the post-revolution French Republic because it was a threat to the monarchies? Such an event is clearly not evidence that republics can't work.
You had two major examples of democratic republics, which were the American revolution and the French Revolution. The American Revolution mainly survived because America was in the middle of nowhere, not see as very important, and the British were tied up fighting the French at the time. Without the American Revolution being a successful example, then "Republics" would have the same track record as what you're saying would happen to anarchism. So, pointing out that what happened to the French Republic might happen to anarchist governments because vested interests might try and overthrow them isn't a specifically good argument that anarchism, in itself, can't work. you're actually arguing why
any new idea can't work. For example, the same argument could be made that Uber wouldn't have worked because taxi companies have the regulators in their pockets and won't let the competition into their cities. Most new ideas are disruptive and vested interests will try and stop them. This isn't an argument that the ideas themselves are good or bad, or will or won't work.
EDIT: also if you look up the CNT in Spain and the model's details, you'll note the term "federalism" turns up quite a bit. It's a form of direct democracy, but you do have larger bodies built up as federations of the smaller bodies. The federalized organization is what oversees the individual companies. As for the "how can the army function?" arguments, well the exact same argument could always be made about any form of government that's any less than a completely militarized police state. You see people making really the same exact argument that if they, for example, abolish the death penalty then murders will run amok. Pretty much take any single step away from a police state, and *someone* will say that's the last straw before chaos.
Anarcho-syndicalists would create a federation of local bodies. Anarcho-syndicalists also believe in organized direct action, that's kind of the guiding principle. There's no reason to think that such an organization isn't going to be able to organize to protect itself and its members / citizens. They're not a "do nothing" type of philosophy. Saying that if anarchists were the dominant political force then everything will be local and there won't be any organization anymore is a straw man argument. You seem stuck on this idea that in this system there will be no higher levels of organization other than local councils, but that's clearly not even a thing. It's not a coherent argument, because it's not what they are saying (hence, a straw man argument).
As for police, you can
totally have local policing. The USA largely relies on city and county-level policing rather than state or federal policing, so there's really no reason to think municipalities run on anarcho-syndicalist direct democracy lines won't be able to police their area. Anarcho-syndicalists believe in organized direct action. if people are fucking other people up in an area they control, you can damn well be they're going to get organized to do something about that. They're not going to print leaflets.