Actually, the FDA does regulate medical procedures. This is to prevent quackery. It's especially relevant with invasive procedures that transfer tissue from one area to another. Not all cell types are harmless in all areas of the body.
Stem cell treatments are something to which I pay close attention (for obvious reasons). There is a lot of evidence that mesenchymal stem cells can promote healing, but that hasn't been proven in humans. There was an clinic in Louisiana called TCA that was doing MSC trials but the FDA shut them down because they were also selling the procedure outside of the study and against the filed protocol. This is no different except the clinic in TFA isn't even bothering with doing real scientific clinical trials. Preclinical animal data is great, but mice ain't men. There is still much work needed to find optimal dosing levels and tease out other potential hazards.