Watching efforts to repeal and replace the Affordable Care Act fall apart was more than just a setback for conservatives who disliked the law, which expanded the government’s role in health care and created an expensive new entitlement program. For some, it was a demoralizing glimpse into a future in which Republicans have all the power in Washington but are powerless to do anything with it.

“Anytime a party is given this kind of opportunity, you’re judged by the product you produce,” said Josh Holmes, a Republican strategist and former aide to Mitch McConnell, the Senate majority leader.

The inability to produce is especially problematic for the Republican Party, which portrays itself as more capable and efficient when it comes to running an unwieldy federal bureaucracy. “Where Republicans have their biggest problem,” Mr. Holmes said, “is when all of a sudden they look like they don’t have their hand on the wheel.”

In that sense, it is competence — and not the accusations of corruption or collusion that have led to various investigations into the Trump campaign’s ties to Russia — that most worries many Republicans.

A perception of ineptitude could be especially damaging for President Trump, who portrays himself as a master problem-solver and deal maker and promised voters that the country under his leadership would be run so competently, “you’re going to be so sick and tired of winning.”

Many conservatives brushed aside doubts about Mr. Trump’s readiness to be president, and his true commitment to conservatism, and voted for him because he represented their best shot at pursuing an agenda that would begin rolling back what they saw as an egregious expansion of government under President Barack Obama.