I think the big distinction is not between Advantage and Supplemental, but between HMO and PPO. So this is not really a Medicare only question.
With HMO, you're covered with their providers, but need special permission to go to other doctors. This may take you months to get, and if you don't live in a big metro area, there may be only one specialist, he or she may have a long waiting list, and when you get there, might not be so competent. Also consider the distance to their contracted hospital, if you're likely to be admitted to one. Kaiser is big in our town, but their nearest hospital, and many of their specialists, are 25 miles away, and there is no good public transport.
With a PPO, you can see a much larger groups of doctors without waiting, although some may have a smaller copay for you than others.
DW was a mental health provider; she had a couple of HMO patients who died because it took them 3 months to get their specialist visit and diagnose (for example) liver cancer.
Depending on where you live, there may be Advantage PPO plans although most are HMO. In my county in SoCal, I have a choice of 32 HMOs and ONE PPO. I've seen other areas that have several, though.