If Matt Gaetz could drag his party to the right, then Jim Jordan would already be Speaker. The reason Jim Jordan isn't Speaker tonight is that some members of the GOP are resisting further moves towards extremism.
If they can ultimately be won over to extremism, then Jordan won't need support from Democrats.
If they prefer bipartisanship to extremism, then they must find a Speaker who will actually work with Democrats.
But bipartisanship means supporting legislation that advances at least some Democratic priorities. "Stopping the rightward swing of the Republican Party" and "making Matt Gaetz less influential in the GOP" is not a Democratic priority. At all. Democrats don't care about internal GOP squabbles. If anything, painting the GOP as extremist would help Democrats in 2024.
Finally, nothing in the House is apolitical. So supporting an "apolitical" candidate doesn't help Democratic priorities either, since an outsider "hero" is powerless to push through any legislation, much less push through something that will help Democrats.