It is not private at all (not even a little bit). And, while I understand the state funding you mention, resident of the district do send the school money. An alternative, as has been mentioned, is school choice where the taxpayers' money can go to the school of choice. I assure you, this bill is far more desirable to local public schools than school choice. Sure, districts will develop local policies regarding the non-enrolled UIL participation but, if I understand the bill correctly, will not be allowed to deny the otherwise eligible participant. Those policies will have to be consistent for and fairly applied to all participants I'm sure. I see this as the best way for public schools to put a lid on the school choice push if they handle it correctly.
I beige you are correct though and that most Superintendant/board teams will resist and the only reason, as you mention is money.
Edit to add:
Being of a small government mindset, I also believe their should be no taxation without representation. This seems like a decent I compromise.