I think a lot of people want a Nate Oats "type" of fiery passionate head coach with proven ability to win conference championships and get to the final four. Not necessarily Oats himself.
I personally would prefer a young, modern coaching style, fresh face with a chip on his shoulder than an established old school guy. A lot of people will throw out like a Josh Shertz or similar. Don't know exactly who I would hire, if I were the AD at this point. Jordan Fee (would never get the UK job right away) but that guy will eventually get a major program job. Young, passionate, just wins basketball games (0.917 win percentage so far) Won the regular season and tournament title, and went to the DII Elite 8 in 23-24 season. Coach of the Year in his conference. Then got hired as assistant HC at Florida Atlantic as he moved up to D1 for the next two years and got hired as Lamar's head coach for this year (D1) and is already off to a 1-0 start. Someone like that (not necessarily him specifically), on the rise might rally the fanbase like Stein has for football.
However, the established, proven big game winner and elite recruiter is probably the ONLY thing this fanbase would accept at this point.