You are CORRECT that they are treated as drugs. However, they are approved through the BLA process and are considered BIOLOGICS. It is a common misunderstanding. Clinical trials are of course required. Typically however adult autologous stem cells have a better safety profile like Myocell and Adipocell have. This is why massive amounts of patients aren't needed for phase 3 clinical trials.
A direct competitor Mesoblast uses allogeneic stem cells. So although they are adult, they aren't autologous and that is why they just started their 1700 patient phase 3.