Bush 41 was certainly one of the old-line crowd, but was willing to work with conservatives. I think Bush 43 was more conservative than his father, but probably still full of old-line tendencies.
But really once 9/11 and the War on Terror happened, Bush 43 became somewhat irrelevant in the struggle between old-line and conservative until the left managed to demonize the war and the party in power. That was when the old-line types really thought they had a solid lock on re-taking the party, and Bush 43 worked with them because he was pretty much out of allies otherwise.
You’re looking at it in intra-party terms and I don’t disagree with your analysis. I was commenting on the effects of Bush’s governance (or lack thereof) on the electorate. I won’t do the laundry list of his blunders, but they were legion, so much so that he destroyed the party’s brand on the national level.