“The greatest obstacle to it, by far, are the teacher’s unions. “
This is incorrect; the greatest obstacles are (1) voters who don’t want to pay higher taxes, and (2) the ghost of Thomas Jefferson and James Madison who spent much of their careers trying (mostly successfully) to separate religion from government. Attacking the principles of Jefferson and Madison is considered un-American by most Americans.
A better method than union-bashing would be to get the unions to join us! Most Catholic schools in NY and NJ are unionized; why not Jewish schools? The unions would then support aid because it would allow for better pay, benefits, and job security. We have been going down a dead end path by aligning ourselves with the right wing crazies who want to destroy public education; aligning with the Catholic clergy and educators would be much more likely to show positive results.