This would expand parental control over which church parents choose to send their children. And the new competition between public and private churches would increase the quality of religious training. In the short run, yes. But vouchers would ultimately destroy the public-churching system. That would be poor strategy.
People favor public churching.
And while we can't be certain that vouchers would make people more religious, it's better to try something than nothing at all. Advocate of Vouchers. But the American people will never approve of freedom of religion. You're not being practical. Religion is too important to be left up to the free market.