You can’t prove crypto is yours either. All you can prove is that you have, or once had, access to it.

Right, and cold wallets are super fucking secure like banking is. Like, the cold wallets have the same level of security as banks (more so depending on which one you use). Its the exchanges you're thinking of (you buy from an exchange and send to your cold wallet, and send back if you want to sell it or xfer from your cold wallet directly to the other person's wallet to pay them). Exchanges get hacked all the time, but you can sue/get recovery depending on the exhange (Robinhood is safe-ish).

That’s far less arbitrary than the backing of some random collection of anonymous people saying “trust me, bro.”

Crypto is decentralized, so that's not really a thing. Unless you mean like the global crypto user population, but at that point you're basically at the same population of some countries, just more spread out. And, at that point, isn't that the same thing as governments and their money?

Because know-your-customer laws are quite effective at finding the proceeds of crime and stopping money-laundering.

Must be nice living in a country that cares about its people. But, even then, there's no reason that an electric company can't set up a unique wallet it controlls for every customer to pay into so the company can know its bills are being paid by monitoring said wallet addresses.

preventing interference in politics by anonymous actors, often hostile states.

Ha! I got a moon to sell you.