Yes, CMU is very much on a different branch of the college family tree.
I’d actually suggest in some notable ways, Pitt seems to me like it might be a better fit. It has a somewhat similar history to William & Mary and the University of Virginia in that it predates land grant universities, and only became “state-related” in 1966. I do think that relationship transformed Pitt more than, say William & Mary, but I also think it still remains a bit of a hybrid between the private research university form, and the typical flagship public university form.
Indeed, size-wise, and in terms of balance of undergrad/grad, it is a reasonably close match to UVA–UVA has around 17000 undergrad, 8000 grad, and Pitt is about 19000 undergrad, 9000 grad. Very different from, say, Penn State, which is more like 40000 undergrad (main campus), 6000 grad.
Anyway, not to put on the hard sell for Pitt, but it has become a very popular “likely” choice in our circles for the sorts of applicants who might have a Yale or Brown as a favorite reach, and schools like William & Mary or UVA as a favorite target.