If the rubber has loosened, you can make a tiny hole in the bubble and inject a glue to re-attach it. If you do not have a syringe of some kind to inject glue, just make a short slit so you can get the glue in and spread it. When finished, seal the hole/slit seam with a bit of caulk.
I would use the slit technique on any areas that feel soft, since that suggests that water has gotten in and rotted or delaminated wood underneath. You need to inspect for damage and repair beofre re-gluing the rubber. You can carefully cut back the rubber to expose as much area as needed. You can get EDPM rubber patches if the original rubber is damaged in the process.
I would not replace the whole roof unless there is major damage underneath or many, many leaks. As long as the rubber is reasonably sound, patch it or even coat it withh a rubber roof coating. Special "paint-on" roof coatings are available for EDPM rubber roofs and they work fine.
It sounds like the previous owner did not use the proper stuff when he painted it. It may have just been a cosmetic job, maybe even regular house or deck paint. EDPM roof coatings won't peel if the roof is cleaned first.
2004 American Tradition; 2014 Buick LaCRosse
Homebase in the Ocala Nat'l Forest near Ocala, FL