I used to live in a state that used to have totally pre-assigned matchups for the first round of the basketball tournament AND pre-assigned sites for every game. The latter was intended to put every game on a neutral court, but it didn't work out that way.
That state had a strictly single-elimination tournament. A team had to win five games to get to State. That state's governing body worked around its neutral court issue in two ways:
1. Instituting a seeding system for the early rounds like Tennessee and a lot of other states. Higher seeds would get home court, but for no more than the first two rounds. After that it was all neutral courts.
2. Asking schools to sign up to be neutral site hosts, creating a pool of geographically-friend places for the governing body to assign games. Nearly every time, games would go to a school that was roughly the same distance from each competing team. It was a fan- and player-friendly move that cut down on travel time.
Schools had to volunteer to be neutral site hosts. Those with better facilities became regular fixtures on the tournament landscape, and I'm sure they made a little money on the deal. The host school would get a (small) cut of the ticket revenue and all of the money from concessions.
Such a system requires participation from just about every member school, and agility by the schools involved as competitors and/or host sites. The governing body would give potential host schools as much lead time as possible; they'd tell a school a week or more in advance that it could be the venue, so it could make the necessary arrangements.
It's a system that works pretty well.