"With sufficient competition (and sufficient money in the vouchers), you should eventually see the schools that cut corners get run out of
business by the schools that hire quality teachers."
For this to happen you'd need offer to vastly outgrow demand, which is not going to happen, neither on public nor -much less, on private money.
No, what you'll get is the rich going to good schools and the poors to bad ones. Hey! isn't this already hapening? No: the poors' ones will be even worse than today since more money will be syphoned out to the riches'.
"I would like to see what would happen if 12 schools all had to put their best foot forward to attract students."
That can never happen on geographically bound businesses. Do you have 12 different ISPs lying down fiber to serve your home now? why do you expect 12 corps building schools within half an hour from your home, then?