If I'm not missing something from the article, this is not for teeth you had and lost, but for teeth that didn't appear to begin with (so you naturally have fewer than normal) due to a genetic condition.
First: the process, as far as I understand it. The hormones involved in growing teeth are hugely complex and have effects all over the body, so this group went after a protein that modulates the effect of those proteins to block tooth growth. They created an antibody that targets the protein, basically unlocking the production of additional sets of teeth.
Obviously there is still a massive gap between this and human trials. First, using antibodies is never without risk because the immune system makes its own decisions and is... schizophrenic. Guillain-Barre occurs when the immune system starts attacking your nerves. If you tell the body that this protein is an enemy, god only knows what could happen- autoimmune disorders, uncontrolled tooth growth, all your teeth just falling out because of cytokines. You could make up potential rare side effects forever.
Second, this required frequent re-administration and immunosupressants. You can't go giving people prednisone for a year just to fix a tooth. It's super dangerous and after only a week various glands are so suppressed they start to atrophy. This actually isn't the worst- as long as you can develop a chemical that binds to the protein like an antibody would without activating the immune system. But, and I'm just guessing here, that's probably really really hard.
Third, teeth are different in humans and rodents. There may be (and probably are!) additional factors in humans- rodent teeth are meant to grow indefinitely, human teeth aren't. The teeth in ferrets weren't very well formed. That may be a deeper problem. Even if it's the simple problem -that dosage needs to be carefully regulated and stable so growth isn't interrupted- the solution to that isn't easy. And surely the dose is very important, as is the localization. You don't want to generate (malformed) teeth to the sides, or god help you multiple teeth in one socket (no clue if this is possible).
Last, supernumary teeth are not usually problem-free. Permanent teeth are larger and there's not much/no room for teeth to develop behind them. If they do develop properly, they may be misaligned due to crowding. You aren't growing any more- I doubt supernumary teeth would come in cleanly even if you removed the old tooth. This isn't a child's mouth any more, and teeth are shaped so that they must come out in a certain order, otherwise you'll need braces. I've no idea if this treatment would cause the root of the old tooth to dissolve (or if permanent teeth can even do that!)- you may need to have it pulled and live like that until the new tooth can grow. That might have negative effects on the socket.
Hell you might even be like me- my teeth are all around two inches long. I look like Venom in an x-ray; needle teeth going from just below my eye socket to just above the bottom of my jaw. I don't know how the hell I grew a second set (plus a few thirds) of teeth. I doubt I'd be a candidate even for a perfected procedure.
> Last, to confirm that USAG-1–neutralizing activity affects BMP signaling to generate a whole tooth in a nonrodent model, we systemically administered antibody #37 to postnatal ferrets that had both deciduous and permanent teeth. We observed supernumerary tooth formation in maxillary incisor like the third dentition, although a five times higher concentration, three administrations of antibody #37, and immunosuppression were required (Fig. 6, A to D). The supernumerary tooth was likely to have a similar shape to the usual permanent incisor, located to the lingual side of permanent teeth, whereas a shorter root seemed to be growing (Fig. 6, E to G). Therefore, this supernumerary incisor might be categorized as the third dentition (32).
If this moves beyond animal tests, the potential is there even for people who got all their teeth.