There are many specialties that cross the spectrum of what we call 'Cosmetic surgery' and some are better at some things than others. Anatomy should cut across all specialties, but keep in mind that knowing anatomy doesn't translate into doing a good job in injecting or numbing.
In fact, you can know a whole lot and do a lousy job of getting a patient numbed. Having said that, I would hope that the more educated a physician, the better job overall they would do.
In our office, we don't even find the need for nerve blocks as I add the injection medicine to the syringe of filler and use a super potent topical numbing medicine along with a very slow and steady technique. Usually, our patients who have done blocks in other offices or ours before we developed this technique ask to NEVER get a block again.
So, you see, there are many things to consider here. One other, by the way, is that you should always insist that the doctor do the injecting. If they are farming it out to a nurse or PA, stay away! Good luck.