Anytime you have a public facility, there has to be some system in place to determine who gets to use it, when, and how much.
I have no idea about your particular situation, but around here there are public facilities that are contracted out to specific youth baseball organizations. The particular organizations assume certain duties of maintenance, and also agree to adhere to certain rules governing parking, limits on nighttime hours, limits on use of lights (which are costly and often complained about by neighboring residents), etc.
Parks & Rec normally handles it, but they can be overridden by the county Board of Supervisors if you want to take it that far. But contracting with non-profit youth baseball organizations is common and it imposes order on what would otherwise be a pretty unruly process.
Also, you have to consider that fields can't be used 24/7 without getting worn out. Mound pits, batter's box pits, paths through the grass, etc. Every field needs to have its use balanced against its need to recover.
We do have certain public fields that can be rented on an ad hoc basis, but those are hard to get because demand for fields usually outstrips supply, and the organizations usually get there first. And, they have a history and a vested interest in keeping up their end of the deal, something the stray travel team doesn't have.