“We have a problem in our conference where a lot of our members fear engagement with the media because of the media bias that we all believe to exist,” said Representative Matt Gaetz, Republican of Florida. “Liz seems to understand the importance of doing a lot of media and also doing hostile media.”

Representative Steve Scalise of Louisiana, the No. 2 House Republican, said Ms. Cheney “hasn’t been afraid to call out some of the most radical members of the socialist Democrats.” But her tendency to name-check her opponents makes at least some colleagues uncomfortable.

“I think we have to get away from personalities,” said Representative Tom Emmer, Republican of Minnesota and the chairman the party’s campaign arm, in June, long before Ms. Cheney’s spat with Mr. Paul. “From a messaging standpoint, I think it’s a mistake — you don’t use names. This is not about the people — this is about their ideas. We need to have a battle of ideas in this country.”

Ms. Cheney’s meteoric rise has injected the politics of the personal into the highest levels of congressional leadership in a way not seen since Newt Gingrich, the former House speaker whose political action committee instructed Republicans to “learn to speak like Newt” by describing Democrats using words like decay, traitors, radical, sick, destroy, pathetic, corrupt and shame.

“I think that she’s been very effective when she’s been on TV,” Mr. Gingrich said in an interview. “I think she is personable, knowledgeable and assertive without being hostile.”

And in a party where 90 percent of House Republicans are white men, Mr. Gingrich said, Ms. Cheney is a huge asset in Republicans’ efforts to demonize three liberal freshman Democrats — Representative Rashida Tlaib of Michigan, Ms. Omar and Ms. Ocasio-Cortez — who have become lightning rods on the right, fueling Republican fund-raising.