Well, I don't want to get in trouble for talking about religion and/or politics, but it's possible for tax monies to be used for non-public schools, but just because something's possible doesn't make it a good idea. There are, however, plenty of non-public monies out there (even if they're not being tapped very well). At the high school level, there's at least one Catholic high school in Houston that offers seriously competitive salaries and benefits.
All that said, I don't regret my decision to work where I'm at. I am doing good work, and I am getting to do my part to help some great kids. I also have some real options for next year should I decide to move to a different school.
For example, there's a charter school in Houston that focuses on a classical education curriculum that would be neat to work at. I've taught in such a school before. It was a hoot teaching Latin and Roman history to grade schoolers.
The key for me is getting to teach. I've tried other work since getting out of the Army many a-year ago, and nothing compares. When I interview at a school, I'm usually asked, "What do you teach?" or "What subjects do you teach?"
My answer: "I don't teach subjects. I teach children."