J&W22:
Whether Kollel drives up tuition prices depends on the concentration of Kollel-type (including Klei Kodesh on discount) in the school you send to. Most OOT schools have big issues with this, to the point where it has already been posted here that the schools are unable to offer discounts to working individuals, due to the large discounts for Klei Kodesh, including those working in the schools (and sister schools).
However, somewhere like Teaneck, where there are fewer Kollel-type, Kollel-type is not the driving force behind large tuitions. There the schools need to “keep up” making the school similar to a prep school, but accepts everyone, driving up costs for those that pay.
A third type is the Lakewood type school, where there is a minimum tuition (since everyone is in Klei Kodesh). There is then an incentive to hold down costs (lakewood is known for smaller tuitions), but it may be to the detriment of the quality of education.