I dunno... I think there's still a "team culture" aspect that can't be overlooked.
Look to the NFL. They're all mercenaries. But there are some rivalries that are real, and it certainly seems like the players care. I think there's a level of team culture that persists in those locker rooms that make rivalries important.
I don't see that NIL / transfer portal can kill that. It might reduce it, to an extent, but I don't think it'll kill it.
I think you are right in a way.
As you guys know, I'm a bit of a history buff and in the reading I've done I've learned that the vast majority of soldiers (same demographic as football players, young men) don't actually fight for ideology so much as 'brotherhood' or basically what you called "team culture".
It is romantic to think of Union Soldiers or WWII American Soldiers risking life and limb "to free the slaves" or "end the Holocaust" and distasteful to think of Confederate and German WWII Soldiers fighting to "perpetuate slavery" or "support fascism" but in reality only a VERY few viewed those wars in those terms. They mostly fought because they were on a team. The view of the average soldier was MUCH narrower than our view looking back. For the most part they weren't looking at the big picture questions, they were looking at being a part of their team (and I don't mean this on a national or even divisional level but more of a squad level) defeating the pillbox in front of them or taking hill #372 or somesuch miniscule piece of the overall situation.
One little note on this:
Admirals and Generals for generations were aware of this at least conceptually and it was one of the reasons that they resisted splitting up the kids from the same hometown. Many of "the brass" thought that soldiers would fight harder if they were in units with their brothers, cousins, neighbors, etc than they would if they were thrown in with a bunch of strangers.
This led to some very unfortunate results. In warfare if one side takes say 10% casualties it isn't usually randomly every 10th guy. If an Admiral sends out 10 ships each with 100 crewmen and one of them gets suck you don't lose 10 guys from each ship, you lose all 100 guys from one ship. Similarly, on land, if a general sends out 1,000 guys to take a given objective and 100 of them are killed it isn't usually 10% of each regiment. Instead it is typically something more like 90% of the one regiment that got cut off and 1% each of the others.
A well known example of the potential results of keeping guys from the same area together is the case of the
Sullivan Brothers. They were five brothers who all served together on the USS Juneau and all died when that ship was sunk in what became known as Ironbottom Sound (due to all the ships sunk there). There are plenty of other examples. Due to the geographic organization of most Civil War units some American towns saw nearly their entire population of young men wiped out while others saw few or no causalities.
So back to football:
I think you are right at least in so far as it concerns the players immediate motivation. It really isn't to win for "the school" so much as to win for "their team", what you called "team culture". That said, however, from that perspective does it really matter who the guys on the opposite sideline are? They could be Wolverines, Boilermakers, Badgers, whatever the only thing that matters is that they are "the other".
Edit to add that I'd be interested to hear
@SFBadger96 's thoughts.