I don’t see the issue as that cut and dried.
I think the question is primarily a philosophical one about public policy - Are we giving money to poor students to get an education, or are we giving money to colleges to educate poor students?
The current Pell program gives grants to poor students, and they can decide which accredited college will give them the best education. The student then pays with their own money plus the Pell grant plus financial aid.
With this philosophical stance, saying that a student can only use their Pell grant at a public college would be strange. It would be like saying that welfare recipients aren’t allowed to shop at the same grocery stores as non-welfare recipients; they should only be allowed to buy food at public institutions like … say, the government surplus cheese store. With this philosophy, only allowing Pell grants to be used at public colleges treats poor students like second class citizens. (Of course, there are almost no public grocery stores and public housing isn’t considered very good, while there are many wonderful public universities.)
An alternative philosophy is the one behind public education. There, the state / local government runs the school and supplies some or all of its budget. The government’s money goes to the school, not the student. The government gets to run the school - they can practice open admissions (K-12) or selective admissions (public universities); they can set tuition ($0 for K-12, $$$$ for public universities); they can control the curriculum (tightly for K-12; loosely for public universities).
With this philosophical stance, the government could abolish the current Pell program and replace it with further subsidies for the tuition of poor students. Poor students can decide whether they want to attend a cheaper public college or a more expensive private college. (One complication is that the Pell program is a Federal program, while it’s the state governments that run the public universities, so I suppose the Federal government might want to take control of public universities so that poor students didn’t have to pay out-of-state tuition unless we also forced poor students to stay in-state.) Remember also that state governments spend a chunk of money running community college systems. Their mission is often focused on educating low income students at low cost.
I guess we could convert systems, but it seems like a lot of work for little-to-no gain to me. I don’t think it’s a national tragedy that a relatively small number of Pell grants are being used to help send poor students to schools like Harvard. Honestly, in my case I thought it was quite a good thing
We means test the students already. How do we means test a school? Does this mean that poor students should only be allowed to attend schools that are near bankruptcy? Public universities like the University of Michigan have operating budgets of billions a year and almost $10 billion of endowment, so it would probably fail a means test. I suppose you could concoct a means test that Michigan passed but Harvard failed, but it seems quite mean to discourage poor students from attending Harvard and Stanford.