So, to be clear, price controls never work. Prices are determined (mostly and usually) by supply and demand. More demand -> higher price. More supply -> lower price.
The price of an education (regardless of cost structures, public funding, or whatever) is going to dominated by demand, since supply is relatively fixed. It takes years or decades to increase the supply of qualified instruction following typical university models.
In this case Federal dollars guarantee a massive surplus of demand, while Federal accreditation limits supply (rightly or wrongly).
"Public" schools are paid by students, just like private, as a fee-for-service product. For most educations at most universities, this service is largely indistinguishable. Some of the University income comes from tax offsets, but the product they provide is the same service and on the same market as private universities - fee-based educational services.
The price is therefor entirely determined by the amount of supply (relatively fixed) and the amount of demand for fee-based education. Demand has risen dramatically since WW2 due to a bevvy of Federal programs designed to underwrite and promote post-secondary education.
No amount of legal wrangling or tax-offsetting will defeat that. In fact, by increasing tax-offsets for public universities, the apparent price of the service supplied by public universities drops (relative to the market price), which is a signal to buyers that they should buy MORE of that service. This signal would naturally increase demand until the price of a public education on the market matches the price of the same education at private institutions.
The actual oddity is not that public education pricing keeps rising to private institutional pricing, but that it is not already at the same price.
Prices drive expense-side efficiency. An operation (such as a University) will not be mechanically driven to keep costs significantly lower than income - most are non-profit. There is no reason to "increase margin". Most Universities are already teaching at capacity. Since the price is set by the market, there is no reason to keep prices significantly below the market rate. Instead, the costs associated with an education simply rise to meet whatever revenue can be generated from a fixed pool of buyers. These buyers have virtually no spending limit, since their purchasing is de facto underwritten by enormous Federal programs. The result year-over-year dramatic price increases.
For extra credit, this is largely the same process that drives healthcare pricing in the US. A relatively fixed supply of a specialized service and a price-insensitive buyer pool largely underwritten by massive Federal programs.