no, the signer's key is only used for commands meant for the signer (i.e. create_account)
I wrote a simplied spec from the one being currently discussed; hopefully that clears it since it doesn't even mention that the signer can have a pubkey of it's own until those commands are introduced
https://gist.github.com/pablof7z/dc5b08c3e39cb73512473a53f5f83b48