"The Republicans like having one of their own in the White House, but they don't really have one of their own. They have Trump."

I'm afraid I have to take issue with that. Donald Trump is nothing if not an opportunist, and when he decided to run for president he made an informed decision that the best way to accomplish his goal was via the Republican party. So he made the switch and the rest is history. While he is certainly not an "establishment Republican," he was certainly popular enough among all Republicans to win. So instead of saying Republicans "have Trump" I think it is more accurate to say that they now own him. He emerged from their ranks, he filled his cabinet with extreme right-wingers, and now he's governing like one. He has, in short, become one of them. 

As for the health care flop, Trump can try to blame Democrats all he wants. Since Republicans own the entire store, this is all on them. It was their failure with Trump leading them over the cliff. Good riddance.