So public district A in big city might have tax dollars coming from a half a million people or more, (Houston, Dallas, etc.), while the private school with 250 student enrolled is funded by 250 students. Buildings and staff are expensive. Band instruments are really expensive. That's why so many private schools only have choirs. Voices are cheap as opposed to a full complement of large brass and woodwind instruments and percussion equipment.
Your schools doesn't have a band budget because they are prioritizing other programs or staff. Public schools are always funded better than private schools. Private schools just tend to prioritize the budget and especially focus on discipline and teaching which too many public schools don't get that option because they teach whoever shows up, baggage and all. And that isn't a bad thing. All kids deserve an opportunity to learn. But sometimes, public schools don't get the option to remove those that choose not to.....
A private school should not be forced to compete against 6A programs if they have 100 kids in the high school and 20 in the band program just because you decided their tuition rate gives them an imaginary advantage. That's ludicrous.