I think the reason is actually a little different. I think the appropriate analogy is to Jesse Jackson, Sharpton, and other professional victimization cheerleaders. If their flock realizes that success is within their reach, based on nothing more than their own skills and hard work, the rabble rousers are out of a job.
Similarly, under a voucher system, teachers would be rewarded for skill at teaching rather than political reliability, union/political involvement, and seniority. The power of the traditional educational establishment to control the content of the kids' "education" (indoctrination) would be at risk. Compare this to the mainstream media whining because people can now get news from the Internet, the content of which the moguls can't contol and sanitize.
FWIW, we put our children in a Christian school this year, I don't like having to pay twice for it, and I could spend the money on other unimportant things (like food/clothing/shelter). It's probably one of the best decisions we ever made. And in this case, I think it's unConstitutional that my children must suffer through a secular lack of education.