“The underlying dynamics that drove out John Boehner remain,” Mr. Dent said of the former speaker. “Now you have Donald Trump and all his rhetoric, and he knows most of his members want him to support Trump while others like me want to keep their distance. Then, not being able to pass a budget may be one of the greatest frustrations for the speaker. A lot of members don’t want anything to do with firearms or the budget. They want to cancel the session after the election. That is where we are.”

Unlike Mr. Boehner, who went sailing into private life after being deposed by conservatives, Mr. Ryan continues to cheerfully promote the House agenda.

He makes frequent television appearances and attends local media events at home in Wisconsin. He will tell anyone who will listen about the conservative policy agenda he intends to pursue next year, in spite of the possibility that Mrs. Clinton might be in the White House. Further, Mr. Trump has shown little interest in his priorities so far.

“When the speaker took the job, he did it to help steady the ship at a difficult moment,” said Brendan Buck, a spokesman for Mr. Ryan. “He continues to try to be a stabilizing figure for the party and this place.”

Also, unlike Mr. Boehner, Mr. Ryan remains largely popular with his fellow House Republicans, who have appreciated his inclusive style. Even among supporters of Mr. Trump, there is respect for Mr. Ryan’s consistency on core party issues.

It is particularly rankling for Mr. Ryan, a former chairman of the House Budget Committee, that he is unable to pass a budget. The most conservative House Republicans do not want to honor a deal struck by Mr. Boehner and the White House last year, which raised statutory budget caps to increase government spending. Many have made it clear they will not vote for a budget with that higher spending level.

This has made the appropriations process equally charged. But that is where Mr. Ryan has other problems. Those bills have failed because of fights over an amendment to bar federal contractors from discriminating on the basis of sexual orientation or gender identity, fights over the use of the Confederate flag and other social issues.