>> Signing with your wallet address proves you are a specific person. So that alone is a great reason to include the wallet.

No, it doesn't. It just proves that you own/hold a private key. You need to build an identity protocol on top of that if you want to prove that you are a specific person/organization. Such protocols have been around for quite some time before crypto currencies. It's just they don't have a good UX, that's why most websites require a password based authentication instead of a key/certificate.

I like a lot JWT/JWT as it has a lot of useful attributes(. e.g the "acr" attribute) so you better start with that than a crypto wallet