But as long as healthcare providers can find someone else to overcharge then they'll do that rather than accepting Medicare patients.
No, Medicare reimbursement rates are about 7% lower (in the aggregate, not individually) than COGS - the marginal costs of providing care. That is, if a test costs a provider $100 to purchase wholesale, Medicare reimburses $93, which doesn't cover the cost of the supplies, let alone covering overhead (wages for staff, office rent, etc.)
Well, independent private practices have been dying out for precisely this reason, and outside of Critical Access regions (which have a higher fee structure), many don't anymore. Some view it as an act of charity (the way lawyers might take on pro bono cases), but the laws regarding insurance segregation have gotten stricter and the logistics of it have become so burdensome that many just view Medicare as too much of a liability. It's one of the reasons Medicare patients have such horrific wait times to see doctors, particularly specialists.
Hospitals take Medicare because many of them are legally required to. If they're part of larger hospital systems, they can also play interesting tricks to manage the respective patient populations without explicitly discriminating based on insurance provider, which has been another major force behind the massive consolidation of hospital systems that we've seen over the last decade.
The other major force, incidentally, is that many of them have been going bankrupt, and are being bought out either by other hospital groups or by insurance companies.