Considering the millions of people struggling to pay for college and desperate for lower tuition, this is totally untrue. This is like saying, "if we didn't have food stamps, food would be half the cost."
I am not going to make the analysis for food stamps. Maybe some stores would have lower prices, no idea, as I don't know the model of issuing food stamps.
Think it like this: Alice comes to your store and says I am going to pay 50% of whatever Bob buys. Bob has spending power x. Now Bob, suddenly can buy 2x. However, Bob can buy that many stereos costing x (wlog) and is probably going to just spend x/2 and let Alice pay the other x/2.
Or you can say, well I am the only store in town selling stereos, so now stereos cost 2x, as I know you can afford them. Bob, as he has to buy the stereo, is going to pay x and the store owner is going to pocket an extra x, as it is a monopoly (or oligopoly) and thus can keep the prices high.
So, if you substitute Bob with an average student and Alice with the government, you will see that the student kept paying the same (at best -- you have to account for market pressures here, general currency inflation etc which cause more inflation) and the government has to keep paying now (or you know break the oligopoly; set different rules).
If we provided people with unlimited special accounts only good to buy food (with money, not food stamps), this would definitely affect prices in groceries.
Plus, the nature of the subsidy is important. State-sponsored universities don't seem any more culpable than private ones in this mess.
Assuming 1 teacher making 100k teaching 20 students * 5 classes * 3 credit hour classes a semester ~= 333$ / credit hour. 128 credit hour degree ~= 43k over 4 years. And that's assuming relatively small average class size and ignoring adjuncts etc.
Yet, many schools are charging 4-10 times that much. The question becomes what are the 3+ other people doing if not teaching and can you get rid of that?
PS: And sure some professors make well over 100k and facilities are not free, but again many people even at expensive schools are pulling in less than that.
Assume a PhD student making a $35k stipend or a foreign grad student making $14/hr.
This covers a good 25%-40% of the curriculum, depending on the institution.