Yes, it is enjoyable to face new and different teams. It is enjoyable to play in new and different venues, to see different stadiums. It is enjoyable to see different parts of the state. The transformation of mystery into discovery is one of the great generators of enjoyment in life. It is one of the reasons people go on vacation; to see and experience new and different things.
When bracketing the NCAA college basketball tournament, the NCAA tries to avoid rematches between conference rivals for as long as possible. That is done partly for this same reason.
If it could, I suspect the DuPage Valley Conference would prefer not having to schedule conference rivals to play what would otherwise be a nonconference football game. It is generally not desirable playing a team for a second time. It would be more fun playing a new and different team. If that were not the case, then Naperville Central could just play Naperville North nine times for its regular season. Scheduling would be a lot easier in that scenario. That is taking the case to its logical extreme, but sometimes that is what it takes to make a subtle point much clearer. Variety is desirable.
The transformation of challenge into accomplishment is a second great generator of enjoyment in life. The challenge for top public teams to play top private teams, and vice versa, is another reason for not having separate playoffs. It is enjoyable to periodically challenge oneself. That is why we play games in the first place.
A third reason is that a 32-team playoff allows many teams to play two games more during the season than an 8-team playoff would allow. Players like to play. At least I know I did way back when I was in high school. I suspect that fact has not changed. I suppose the public schools could still construct a series of 32-team brackets if they reduced the number of playoff brackets from eight to seven, but the private schools could not reasonably do so. The private schools would either need to have schools as small as Ottawa Marquette playing schools as large as Loyola, or, apparently like Tennessee, they would need to allow 2-7 and 3-6 teams to qualify for the playoffs. Neither of those two circumstances seems desirable.
So, you might ask, why should the public schools care what happens to the private schools? It is simple common courtesy to show a consideration for others. In addition to reading, writing and arithmetic, perhaps our society would be better off if we also instructed our students in the precepts of morality and good manners. We seem to have taken a large step backwards in that regard over the last half century. This might be one area where our schools, by example, could teach their students a valuable lesson.
So, while avoiding a playoff that is an unnecessary repetition of the regular season is one reason to not separate the private schools from the public schools, as you can see, it is not the only reason. In summary, the playoffs are more enjoyable for all concerned when the public schools and private schools play together.
