For instance, the third party gives you a nonce, you sign it with your private key, and they verify it with your public key.
Presumably, there's a commonly-trusted authority that signed your public key, attesting that the key corresponds to some "identity".
If you're concerned about revealing your identity to the third party, there are other schemes. Check out https://datatracker.ietf.org/wg/privacypass/about/
Game companies won't accept a system that has a non-recoverable state like a deleted private key.
If you can assign a new key to your account, then the private keys don't improve trust, and remove the point in having them instead of just OAuth.
In games like Dota 2, there's already an industry for account selling.
It's pretty simple really,
Find some Boomer's ID, who barely does anything but browsers facebook (or better yet, buy an ID of one of the billion in the 3rd world).
Sign up to this service using their id, hire some poor kids in the 3rd world to level up the account without cheats to build rep, then sell the account to a hacker.
Private keys don't solve any of this.
The alternative is having to run third party closed source anti-cheat software on my computer that is written by god knows who, that has the potential to do an enormous amount of damage if a bad actor were to sneak in a backdoor.
If you look at how much how much time and effort and money people put into these games, it doesn't seem so unreasonable. Another commenter pointed out that you have to give a similar amount of personal information to get a car loan, or, as you say, to get a plane ticket. I suspect a lot of people spend comparable amounts of money and effort on gaming, and probably see it as a bigger part of their lives.
I don't think the idea has merit. See how long it is taking the Real ID act to become enforced. People barely trust the government with this info, with the ability to get on a plane (among other things) at stake.