I thought we were talking about puppets and appearing alone in hearings.
If your premise is that appearing with Cheney during the hearings indicates Bush was a puppet, than by your standard Clinton appearing at the hearings with both White House counsel Bruce Lindsey and his National Security Advisor Sandy "What have I got in my pocket?" Berger makes him one, too.
If the former indicates lack of independence, so does the latter, and even more so. Clinton had two handlers, and by your own argument, wasn't even on the hot seat at the time.
Or maybe you can just drop the whole "puppet" routine for the truckload of horseshit that it's always been.