It should be the other way around. Gambling sites should mention where their companies are registered and provide the reference numbers for that. They should show who runs the show so things are more transparent for the gamblers.
Correct, there would be no player on a certain gambling site if they are going to ask proofs that a certain player is legit, because just like what you have said, it should be the other way around. Gambling sites are the ones who are attracting players, so its their job to do such thing like proving they are legit and people could trust them by providing proofs that they pay, something like this.