Exposed to a national spotlight from a young age, Mr. Bush has been hearing about his bright political future for decades. “The Republican convention is doubling as a dress rehearsal for a man Republicans talk about as an up-and-coming heir to the Bush legacy,” The Baltimore Sun wrote of him in 2000, referring to him as a “hunk” who could put “the passion in compassionate conservatism.”

But that is not the message Republicans want to hear now, Texas political consultants, donors and observers said.

“Everything was lining up to give him the brass ring, but the party changed too much,” said Brandon Rottinghaus, a professor of political science at the University of Houston. “The Republican base changed in such a fast way that many were left without a chair when the music stopped. Bush is a great example of that.”

Jay Zeidman, a longtime friend of Mr. Bush’s, said he believed that those shifts masked a dissatisfaction with the direction the party had taken. “There’s a lack of political courage in this state right now because of Donald Trump,” he said. “I think Americans and Texans are thirsty for some reversion back to what politics used to be.”