Asked this exact question to each person I interacted with: Dean Shelton at the info session, and both my interviewers. Best I can tell is that core clerkship will be exclusively done in HCA hospitals, but preceptorships in years 1/2 are actually rarely HCA and rotations in year 4 would depend heavily on what you are pursuing.
As for the behavior of HCA, I got answers (from Belmont faculty of course, so take it with a grain of salt) that I should think of HCA as more of a franchise situation... Some are well managed and good places to work, some are not. I've seen this view echoed on r/residency as well in my limited research.
2 things to keep in mind about HCA: 1. it's a very large for profit company (#65ish on fortune 500). And 2. It has recieved a lot of heat for over saturating the market with residents, therefore driving resident wages and working conditions down. They have been seen as doing this to help their own profit margin (why pay an attending when you can pay a resident to do much of the work in your hospital?) Some people think companies like HCA are the villains who are contributing further to the broken US Healthcare system.
For what its worth, I asked my interviewers about this and they gave me very transparent answers that made me feel that I could succeed here even with all this negativity swirling around HCA. Ultimately it's up to you to decide if you'd be willing to potentially work at a place with this reputation (at least for a year or 2). Also remember that whole FCOM, while heavily integrated with HCA, is an independent institution. You won't be going to HCA medical school. If you see something you don't like, you don't have to emulate it in your career. Hopefully this has shed some light on the issue as it has also been my #1 concern at this school. Feel free to DM if anyone wants to discuss it further.