It can't be proof of ownership if it's all of those things because ownership is decided by eg judges and not what's written in a random database somewhere.

Contract law. If we agree that what is scrawled on a napkin is a contract, it is.

As we've seen with the reactions of prominent crypto figures who've gotten robbed, approximately nobody really believes that the ledger is the final determiner of ownership. They don't say, "Well, other people own that money now," and move on. They tend get mad and call the cops, just like everybody else.

And as we're seeing in the great raft of crypto-related bankruptcies, judges don't take it seriously either. Just because somebody happens to hold the keys for tokens means very little to them. Something that appears quite palatable to the people who might get some of their investment back.

So in practice, it looks like digital ledgers are more of a "fantasy football" version of ownership tracking: taken very seriously by people who play the game while they're playing it, but not by anybody else, and also not when the game's over.

Contracts can be written on napkins but they're not suicide pacts and judges can edit them after the fact.