The Tor user may pay the fee from a bitcoin exchange account. As far as I'm aware, exchanges do not offer their customers the option of signing messages.

Also, if the Tor user's non-exchange wallet has many inputs to many addresses, and pays the fee from that wallet, which address(es) would the Tor user then have to use to sign the message? And if the Tor user pays the fee from non-P2PKH addresses (e.g., segwit P2SH addresses or multisig P2SH addresses), the Tor user can't sign the message using those addresses.

And again, not every wallet has the ability to sign messages, with mobile and web wallets being the most obvious examples.