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.
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.