You really can’t without logging into it. Like Admin suggested though you can report the site as abuse and have it taken down that way, giving you control over your personal information again.
The only way you can prove ownership of the account is by recovering the password or access to the email address. That’s it, those are the options. If you cannot do either of those things, you cannot prove ownership of the account.
It’s frustrating to lose access to an account like this. But you have to understand that we cannot have an endless list of alternative verification options with increasingly flimsier evidence and subjective calls.
We have to find a balance between making sure you can access your account, others cannot access your account, and protecting your privacy. If we need to improve the first point, that means we need to concede on the second or third points.