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?