But as Mr. Renacci sees it, after helping to send Mr. Trump to the White House, Ohio voters now have a chance to remove one of the impediments standing in the president’s way by denying Mr. Brown, a Democrat, from serving a third term.

“The reason why the president endorsed me and asked me to run is he wants somebody who can help him get his agenda done, not somebody who’s going to vote against him,” Mr. Renacci said as he sat in a booth at a chicken-wing restaurant here.

Mr. Renacci, with an estimated net worth of at least $34 million, according to Roll Call, has had a long and varied business career, with involvement in nursing homes, Harley-Davidson dealerships, a Chevrolet dealership and partial ownership of an arena football team. He has lent $4 million to his Senate campaign. A quiet, casual demeanor helped ingratiate him with working-class voters in his House district south of Cleveland even as he held on to the traditional Chamber of Commerce Republican vote.

“Businessmen and businesswomen have different styles,” he conceded. “I’m more the quiet style.”

All of that makes his embrace of Mr. Trump an imperfect fit. For one thing, Mr. Trump pledged to “drain the swamp,” and Mr. Renacci, who was first elected to the House nearly eight years ago, is testing just how long someone can be in Washington before becoming part of what must be drained.