It sounds like you're in a good spot where the taxes extracted from you go to support a value system that you are in favor of. Given that taxes aren't a choice, and given that value systems vary between people, can you envision a case where somebody with a different value system would be unhappy about where their taxes go?
Sure. Lots of people have ideas or preferences I think are terrible. I hope they lose at the ballot box.
[EDIT] To make this more substantive: I think there's a distinct difference between collectively funding a public system, and redistributing money to folks to go toward private entities—this goes beyond not liking the values or environments of some of the schools private-school-preferrers might spend their vouchers on. That's why I suggested that if we must do vouchers, it ought to be limited to what a given person paid in themselves, not other people's money—I think there's a fundamental difference between funding a public good available to all and giving people cash so they can get past a private paywall to access a similar good.
