In both of those games, Alabama was in prime position to win the game. Had it not been for mental errors + poor play falls (Bill O'Brien in 2022 vs UT), Alabama is in better position to win the game. Alabama didn't lose to Vanderbilt, OU, or Knoxville because of the environment.
I'm not saying homefield/homecourt advantage doesn't have an effect either But the importance fans put on it is greatly exaggerated. Maybe in certain moments it has a greater impact - for instance a last second FG for a kicker - but overall, after adrenaline wears off after a few plays, it's just crowd noise. 90% of what we think plays into homefield advantage just comes from preparation, in game adjustments, good gameplanning, and players stepping up.
LSU was rocking at the start of the game vs Alabama last season and all it resulted in was Alabama running through them for 60 minutes. I'd argue that the best atmosphere in College Basketball last season was Kentucky hosting Arkansas for Calipari's return. Every ESPN commentator and UK fan talked ad nauseum about the environment Rupp would create for his return. And they weren't wrong. It was an electric atmosphere. Result? Arkansas walking out with a win.
The one argument I'd lean towards is that players feel a sense of pride when playing at home. You don't want the opposition to come in and you not defend your home.