IIRC, Medicare Advantage are the "private insurance plans" option.
Once you go on those, you cannot go back w/o risks of substantial increases in supplemental plan costs, and you can lose your "pre-existing conditions" waivers for the supplemental plans for a time period.
As I understand it, the supplementals cannot deny coverage when you first join Medicare, but if you go on Advantage you can be denied those coverages as that waiver expires - and you can have 6 month waiting periods. (Google 'MediGap Trap') You will get the best rates on the MediGap when you first enroll - prices can bump significantly w/ pre-existing conditions if you try to get in them after being on Medicare Advantage.
I've got some years before Medicare, but pay attention to how some of it works so I'm prepared once I need to make that transition.