That seems like it should work, but in the political reality we are currently living in - the Dems have been given the gift of letting the Repubs show that they cannot rule/legislate when in the majority. This process, when it works, allows the majority party, through their conference, to elect leaders of the house and of committees, but it is a crappy system if there is a coalition. And that is what the Repubs are in actuality, seems like this could fracture the party, and it may cause moderate (some might say reasonable) Repubs to actually reach out to the Dems; the problem is that they might want a few to reach over and support Jeffries. In reality, if there was a split it would take a great deal of compromise to get anything passed.
But it would be a huge defeat for the current majority to have to make a coalition; not sure how much worse that could be (or be perceived to be) that the shitstorm the R conference has right now (quick question bigger mess - the 2024 PAC 12 conference, or the Republican conference/)
footnote; Zero democrats voted no in the Republican conference last week that pushed Gym (was Gym one of the lolllipop kids?) out of contention for the speaker.