No, that's not how PhD funding generally works in the US. A PhD student is typically funded either directly by the department of via grants that faculty within the department have obtained (or a bit of both). Graduate students can apply for external funding too, but it's generally pretty competitive, and the money goes primarily to the student.
I consider people receiving money in your name to be paid by that person. I know the departments grants aren't by name, they ate by head, but it's still the same thing.
The college isn't doing any charity here. They are getting paid your "tuition" and they need to take in grad students to keep getting those tuition grants.
I’m not following you. Are you somehow under the impression that departments receive grant money based on how many PhD students they have? It doesn’t work like that.
