A lot of the time, private keys will get stolen contemptuously with an account getting hacked so this wont necessarily solve anything.
Also, the administration will look at a lot more than the signed message when considering to recover an account.
Such as what?
If someone delivers couple of signed messages with a couple of BTC addresses posted here, what other proof could you need? If that doesn't cut it then nothing would.
Like I said, most people getting their accounts stolen is due their emails getting hacked. Getting your private keys stolen is much harder if you are keeping them safe.
The admins will check to make sure the person making the request is the same person who owns the account. The forum does not broker account sales, but will not knowingly help a scammer recover an account they have sold. The admins will review the geo-location, ISP, browser, OS, etc., of the person who is believed to own the account verses the person who is making the password reset request.