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.
I don't think geo-location, ISP, browser, OS, etc is relevant in 2017. Most people use Tor or VPN to access this website, hopefully with javascript off too because you don't want to get hacked by some scammer sending you a PM that has a javascript exploit of sorts which would get your IP, or a simple image hosted by an attacker that opens itself in a PM and reveals your IP could do it. It's just dump to visit a bitcoin forum without some protection. So both good people and bad people are using these things.
Theymos specifically asks for BTC signed messages because it's the only 100% guaranteed way to prove you own the account (getting email, account and entire BTC wallet hacked is too unlikely specially for people that has been here for years, so Hero Members and up)