Despite the turmoil, the House impeached President Bill Clinton in December over the objections of a majority of the public. At the time, many observers thought the backlash against the impeachment proceedings and the perceived extremism of congressional Republicans would damage the G.O.P. in the 2000 election. Expectations about whether Mr. Hastert could manage to unite his “badly divided” and “split, often combative party” were similarly pessimistic.

But Republicans suffered few ill effects despite trying and failing to remove President Clinton from office. The party quickly fell back into line behind George W. Bush, who significantly outperformed political science forecasts and won the presidency in 2000, an election in which the G.O.P. also maintained control of both houses of Congress. (For his part, Mr. Hastert eventually served as Speaker for longer than any other Republican in American history while maintaining high levels of unity in Congress.)

The reality is that few Americans pay close attention to Congress — many cannot even name the Speaker of the House — and most legislators represent safe seats. Moreover, presidential elections are largely determined by other factors such as the state of the economy. Republicans certainly need to get their house in order, but the reasons to do so have less to do with the 2016 election than with the need to maintain order in a chamber they ostensibly control.