FWIW, I don't think GHWB was particularly "normal" as a Republican. He was part of the Reagan cabal that set the party on a course toward the extreme right, for cry eye. The only thing Bush had going for him that made him seem statesmanlike was that he successfully prosecuted a war against Saddam (whom, as you may remember, he successfully rechristened as "Sodom".) Plus, Bush was genuinely avuncular as well, which you don't see much of in today's R.
I think the last of the "normal" Republicans was Eisenhower, and that's because he was basically the last of the extinct Dewey faction of the party.