The only way public schools can be held accountable is if people who send their children to private schools and/or homeschool were to be allowed to deduct this tuition from their local property tax. If the tuition is more than the property tax then people wouldn't pay any property tax.
Public schools would either improve overnight or shut down due to lack of funding. As for vouchers... while I do support the idea of school choice I am worried that government would still be able to exercise control over the schools that receive funding via vouchers. That's one reason to take the middleman out of the picture and have people pay private school tuition directly.
Of course, this idea completely contradicts my belief that personal property taxes should be outlawed by a constitutional amendment.
While I agree with you in principle in regard to property taxes, on must remember those taxes are not just for the funding of public schools. Property taxes also go toward county and/or municipal services which do need to be paid for in some manner.