WASHINGTON — Speaker Kevin McCarthy of California won his job by bowing to the demands of a group of far-right Republican dissidents to overhaul how the House operates, effectively diluting his own power while increasing theirs.

The protracted floor fight that finally led to his election early on Saturday morning made for a historic and tumultuous start to the 118th Congress, but it was easy to lose sight of why any of it really mattered, beyond the captivating personal and political drama of the week.

Democrats, after all, still control the Senate and the White House. The hard-right group that resisted Mr. McCarthy’s ascent only makes up about 10 percent of the House Republican conference. How big of a problem could the G.O.P. disarray in one chamber of Congress be?

In fact, some of the concessions Mr. McCarthy agreed to would make the practical business of running the House next to impossible. It could be left unable to do basic things like fund the government or finance the federal debt. For the dissidents, that was the point. For the country, it could lead to some grim consequences.

The speaker, on behalf of the majority, controls the House floor. The main tool for doing so is the Rules Committee, which sets the terms of legislative debate, including which bills can be considered, for how long and — crucially — what amendments may be offered and by whom.