“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.