Absolutely agree. Markets are great when they meet the conditions for proper competition; and privatisations frequently occurs in sectors that don't meet those conditions; e.g railways.
Edit: in my example the university could change from accepting all students with correct diploma to doing admission tests to resolve. I know nothing of the feasibility of this though.
Then people will choose their high school depending on whether they need that extra preparation, and those schools will have more students or be able to charge more.