Indeed, members of the Freedom Caucus — which is supported by outside conservative groups — have often claimed the mantle of pure conservatism, but their tactics have been seen by many in their party as uniformly counterproductive.

Time after time, they undermined Republican leaders’ efforts to secure wins for the conservative cause by overreaching and demanding the impossible.

They have anointed Representative Mark Meadows of North Carolina as their leader and principal spokesman. But they occasionally get help from three senators sympathetic to their cause: Ted Cruz of Texas, Rand Paul of Kentucky and Mike Lee of Utah.

At critical times, House conservatives have forced their party to make deals with Representative Nancy Pelosi of California, the Democratic leader — an idea anathema to most Republicans, who would seethe.

This time, by negotiating with Mr. Trump on the complex issue of health care, a measure whose complexity he did not seem to fully grasp, they moved an already contentious bill further and further to the right, eliminating too many benefits to keep moderate and other conservative members on board. Even Mr. Trump was said to be taken aback by their attempts to remove things Republicans have long promised to keep, like health insurance benefits for children up to 26 years old.

Given that the House and Senate majorities were built on a promise to repeal and replace Mr. Obama’s signature health care law, many lawmakers fear electoral repercussions. “It is painfully ironic that members from safe, conservative congressional districts who can’t ever quite seem to get to ‘yes’ make it harder to enact good, conservative public policy like repealing Obamacare,” said Michael Steel, a Republican strategist and former aide to Mr. Boehner.

This seemed lost on many Freedom Caucus members on Friday. “Part of the legislative process is working with people who have different ideas,” said Representative Justin Amash, Republican of Michigan. “This process from the beginning wasn’t designed to do that.”