A little perspective about the politics here - people are lumping together all the Republicans, but there are 2 distinct factions.
Pillen is not a Trump guy. Herbster was the Trump guy, Pillen was the establishment R guy in the primary. He's generally the same type as Ricketts - nominally conservative, not an ideologue, generally safer, stays in line with the normies in the state, mostly in control of the state party. Pillen was the more liberal, and his being against so-called "DEI" is not him launching a crusade, it's taking the majority opinion in Nebraska (and much of the country, outside of academia).
The Trump wing is not in love with Pillen. They see him as mostly part of the establishment, and would have issues with a lot of his picks as being personal loyalists & swamp creatures, not reliable & true believers.
The libs on the BoR are probably opposing anything from Pillen regardless because it's all more conservative than they are, but it's distinctly possible that the deadlock is because there are also Trump-wing regents wanting to stop what they'd view as a buddy pick/machine politics appointment.
There's more nuance even within that, but the media narrative about the fault lines is mostly garbage.