I never heard of a grad student program that pays you to do your research project. The ones I've seen, always offer a choice between paying you to work on someone else's research project, or you get to work on your project but you don't get paid; you do get access to facilities, but you have to pay an exorbitant fee for this. Am I missing something?
That's the opposite of my experience and understanding as a fresh PhD. In my field, social science, PhD students 100% own their research agenda. We get paid via a combination of grant (i.e. free money) and RA / TA work (i.e. money you work for). The better the program, the more of the money is free. And you always have the option to apply for external grants to fund yourself. In my (very fortunate) case, I rarely work on others' projects for more than 5 hours a week.
My friends in the natural sciences are more tied to the projects of their supervisors. But in the end they are still expected to be a mature researcher with their own agenda.
Caveat: I'm only certain that this is true for top-50 programs.
That's why I said you have to find a lab that does what you're interested in. I have done just that, and for the last 5 years as a phd student I've been paid mostly to do what I would do as a hobby otherwise (read/implement deep learning papers), and try to advance state of the art.
from my own experience, this is the exception, though. even in CS, if you pick some of the more theoretical problems or a not-so-hot area, you might not get funding - (almost) no matter how good you are. and that's CS. move to other subjects like sociology, linguistics, history, and you will find that barely anyone gets funded.
I haven't heard about anyone in any PhD program in US who's not funded one way or another. Sure, you might need to TA a class, but it's usually not that hard, and leaves plenty of time to do research.
