First, tuition and costs have both gone up for universities, and you can make up whatever costs you want. New sports facility? Part of the costs. 7-figure salaries for administrators? Part of the costs. How would you decide which costs are "reasonable" and which aren't? It's a real morass -- what you are suggesting, to be effective, would require government micromanagement of private education.
Second, government already runs public education -- those institutions cost less, but costs have still gone up and tuition has still skyrocketed.
Third, despite the fact that Medicaid does impose price caps, both the list price and costs for medical care have also skyrocketed.
Explain again how price caps solve anything? Every market with price manipulation looks like a disaster. Housing (rent control) and medical care (medicaid & insurance payments are limited or contractually negotiated) are just two examples.
I think what we want are not mandated prices, but mandated transparency in value, so that you know if the price is worth it!