Well really it was not Pitt that built up the medical center. That was driven internally and financially by UPMC. Pitt had little to do with that financially other than the 3 men who conceived the idea worked out of Western Psych and got the opportunity to in essence take over Presby when they helped discover an embezzlement scheme.
As you note, UPMC supports the university's health sciences to the tune of $200 million annually, not vice versa. UPMC in all practicality drives both the clinical and research operations. The research funds may be housed at Pitt but they are more along for the ride than driving the boat. Not saying that is a bad thing - it is a symbiotic relationship, and the University does its part, but the financial aspect and main decision making is controlled by the UPMC side
UPMC is entirely a university spin-off entity. It really originated from the university's efforts to create an academic medical center which began when Pitt's medical school built Falk Clinic and enticed Presby, Children's, Eye & Ear, Magee, and Womens hopitals to move on campus to form the University Medical Center in the 1930s. It added pieces to this ever tightening integration like Western Psych and Salk to became the University of Pittsburgh Health Center in the 50s, and then incorporated into UHCP in the 60s, followed by a consolidation into what was termed the Medical and Health Care Division (MHCD) of the University in the 80s. And yes, Presby University Hospital was part of this but still operated independently. By this time the University of Pittsburgh Physicians (UPP) was the component of the university that served as the clinical practice plan for all University faculty which had clinical service appointments on top of the academic and research duties that were performed under various other University departments or centers. After Montefiore was added in 1990, the MHCD was renamed to the "University of Pittsburgh Medical Center," a name an acronym whose copyright, by the way, is still owned by the University. UPMC wasn't spun off as a legally independent entity from the University for another eight years, at which time UPP was also transferred to UPMC so the University could divest of its faculty's clinical operations. However, the University retained all operations on the academic and research side. To this day though, health science faculty are employed and overseen by the University for all academic and research endeavors, but their clinical appointment and compensation for clinical responsibilities is handled by UPP, which is now entirely under UPMC. There are arrangements between Pitt and UPMC for a common paymaster for these individuals for employee tax calculation purposes. In any case, the spin-off of UPMC in 1998 was done to shield the University from financial risks involved in being a clinical provider and to allow the Medical Center to be more aggressive in its early battles with AHREF and Highmark (it had already formed Tri-State as a confederation of hospitals like Passavant and Shadyside as a foil against Highmark's monopoly; those obviously later integrated to form the beginnings of the wider health system) which included considerable risks including starting its own health insurance company.
All actual bio and health science research, from the very beginning, has been entirely financed through, managed, and originated from within the University. It is university academic faculty, departments, and centers that are entirely responsible for originating the ideas and breakthroughs that get published and bring in research funding. Without that academic component, none of the top clinicians at UPMC's flagship facilities would have been there. Top medical experts, like Salk, Stazl, Jose-Alain Sahel, and Patrick Moore, aren't landing somewhere where they don't have academic appointments or are able to continue their research in a meaningful way with a robust collegial setting with the appropriate infrastructure and administrative support. Most physicians at major academic medical centers are involved in at least some research and training. That is the prestige that you mentioned before, and that mostly the University, because prestige within academic circles emanates almost entirely from research endeavors which is 100% under the University, conducted by University faculty and staff, and it always has been. And as a corollary, the medical tourism and exportation of the UPMC brand is largely based on that prestige. Otherwise, UPMC is just another Kaiser Permanente or LifePoint in the business of just getting bigger, and not trying to compete with Partners, Mayo, MD Anderson, Hopkins, and Cleveland. Go to NIH RePORT and compare how much federal bioscience research funding has ever been, and is currently awarded to, UPMC vs Pitt. But prestige that UPMC brings, and it does because it is the seat of the clinical aspect, and the prestige that the University brings for research, is symbiotic.
And as I noted, since its spinoff from the University, UPMC's articles of incorporation specifically names two, and only two, entities that is its mission to support: the University and Presby, and that is contractual to the University as well. That comes into a tune of ~$200m a year for the University based on UPMC financial performance, which is about 1/3 of what the University obtains externally from its research bioscience endeavors, and that is decidedly not just being "along for the ride," either in historical or contemporary reality. UPMC has financially supported and partnered with Pitt on many research projects since it has spun off, because it is in both entities interests to do so, they are historically and contractually tied together including co-employment of their top clinical and medical science talent, and it is actually UPMC's founding mission to do so. In these areas, the University and UPMC are of one mind.