And that's assuming your argument is true, which in general it isn't. If you raise prices then some of your customers will go elsewhere. That's why sellers don't raise prices already. If you raise prices and your competitors don't, even if your customers have extra money, people can still count and a dollar is still a dollar. The profit-maximizing strategy isn't to raise prices, it's to expand capacity as quickly as possible (which in many cases is immediate) so that you can make your existing profit across more customers who can now afford your product, without losing them to competitors with lower prices.
Is it the case that existing private schools have much excess capacity? Frequently the argument for private schools is smaller class sizes; a rapid influx of new students without new, excellent teachers defeats the purpose. Presumably new classrooms would need to be constructed and so on.
If you receive a $5,000 voucher and your private school jacks up tuition by let's say $1,000, it is unlikely many parents are going to disrupt their routines and children's routines to reclaim that amount, especially since they are feeling relief from a net $4K tuition break. Tuition has effectively decreased, so why rock the boat?
Suppose you're right and a $5000 voucher will cause a $1000 increase in tuition for three years, after which new schools will open and the competition causes tuition to fall again.
So as a parent, for the first three years I'm up net $4000/year and $1000/year is going to build new school capacity to meet increased demand, and thereafter I get the whole $5000/year. Am I supposed to be unhappy about this?
Or from another angle- what will prevent an ever increasing amount of tuition such as happened with the higher education system? "We need to raise the voucher to $10,000 now…"
And what you're getting at is one of the reasons that a UBI is better than vouchers.
Suppose you create a $5000 education voucher and there was previously a school with a tuition of $4000. That is now over; they might as well charge $5000. And if there is a higher quality school charging $10,000 and people whine that the voucher doesn't cover it and have the value of the voucher increased, now every school will charge $10,000, because that is the only way to get the whole voucher and there is no reason not to.
But suppose you replace that with $10,000/year in cash like a UBI. Can all schools now raise tuition to the entire amount? Obviously not, because all else equal people will still prefer the $7500/year school so they can spend the remaining $2500 on whatever they want.