If you want to call yourself an IVF specialist , then I think you should only do IVF , and nothing else. This allows you to become extremely good – and gives your patients a lot more confidence in your skills and expertise. Good IVF doctors are so busy doing only IVF, that they really do not have the time, energy or inclination to do anything else . And IVF is so rewarding, that there's really no reason to do anything else if you do it well !
Now every IVF specialist has trained as an obstetrician and gynecologist ; and has the skills to do hysterectomies and caesarean sections, but that doesn't mean they should continue doing these procedures.
If I were a patient. the fact that the doctor does only IVF would be very high on my list of criteria for whom to select my doctor. ( Would you go to a cardiac surgeon who also does appendectomies ?) The fact that he does only IVF means that he is better at it as compared to a doctor who does lots of other procedures as well, because he is devoting all his time and energy towards enhancing his IVF skills .
Similarly, I would select an IVF doctor who practices only in one place , and does not run around from city to city or country to country , trying to get more patients just in order to increase his income . Anyone who travels so much dilutes the amount of time and energy he can devote to his patients within one center. I want a doctor who focuses all his energy on getting me pregnant ! If he travels, he is forced to delegate the treatment to his assistants, who obviously aren't going to be as good as the main doctor ! ( After all, an assistant would not remain an assistant for very long if they were as good as the main doctor – they would start their own clinic !)