Most physics PhD candidates do not have free choice of research questions (or, at best, they are spending their "10% time" on their own questions, which is what I did).
On institutions: I did my PhD at the University of New Mexico. A fine institution, but also not Harvard / MIT / etc. I went there because Carl Caves had created a great environment to work on quantum information. Carl encouraged me to work on whatever problems I wanted, but was also happy to provide advice upon request. My fellow PhD students enjoyed similar freedom, and I believe it benefited them enormously as well.
Edit: To clarify my earlier comment: I am certainly _not_ saying that it is universally the case that people are able to choose their PhD project. That's not true, especially in fields involving expensive experimental equipment. I'm saying that someone not able to choose their project is being badly trained. It should count as a very heavy mark against doing a PhD when the advisor believes they should control the direction of the project.
The cost/benefit tradeoffs are atrkly different.
The reason is that choosing what problems to work on is a fundamental research skill. Someone who doesn't learn that skill hasn't really learnt to do research.
This isn't to say that a student shouldn't spend time talking with their advisor about choice of problem. Of course they should -- they should talk to as many people as possible! But the responsibility for the choice of problem should be theirs. If that's not the case, then they're missing much of what it is to do research.
To quote a letter of Feynman to a former student of his ( http://genius.cat-v.org/richard-feynman/writtings/letters/pr... )
"I had another Ph.D. Student (Albert Hibbs) [who worked on] how it is that the winds build up waves blowing over water in the sea. I accepted him as a student because he came to me with the problem he wanted to solve. With you I made a mistake, I gave you the problem instead of letting you find your own; and left you with a wrong idea of what is interesting or pleasant or important to work on..."
[1] This is actually pretty complicated. Many students don't have ideas they're confident pursuing. Some don't grow into it, even after gaining some experience solving research problems. That presents challenges. And sometimes students badly misjudge what is important or interesting to work on. Should a supervisor interfere when that's happening? It depends. Making mistakes in choosing which problem to work on is an important part of learning to do research. So some floundering around is fine. But allowing it to go on for years and years is just cruel.