Just to clarify, the OP will not be doing an OB fellowship in the hopes of finding another route to the OR.
After ~six months of residency in a small community hospital unopposed FM residency, I have to disagree with SansaStarkMD's views on FM training. I was also attracted by the possibility of working in the clinic, hospitalist shifts, small ER's, possibility of OB, etc. The reality is, fewer and fewer docs are doing this. It is exceedingly difficult to be trained well enough in all these areas in 3 years to be competent across all those domains. Even getting enough c-sections around all of your primary training is very difficult, let alone somehow getting enough OR time to gain broader surgical competence.
The push toward further specialization is not entirely political. There is a reason that entire residencies are devoted to OB/Gyn, ER, etc. I would strongly encourage you to consider choosing a residency if you will not be happy with outpatient family medicine. You will find that it is possible to do other things with FM training, though there is significant resistance from hospitals and other doctors. This may seem trivial now, but it is difficult to be in a position where you are treated poorly by your peers who have more training than you. Furthermore, there is a strong push by health systems to be time-efficient and meet quality measures, which is extremely difficult if you are trying to work in multiple arenas simultaneously. This is made more difficult by the increasing administrative burden of working with different EMR's.
You will not have a hard time finding FM doctors and residents who will validate your idea of what family medicine can be. I would challenge you to find recent grads who are doing exactly what you imagine yourself doing. Things in medicine are changing in ways that put both official and unofficial barriers to what can be done by family medicine doctors. I convinced myself that family medicine is something it is not. There are a lot of people who really enjoy it and make a huge impact in people's lives. You might be one of those. However, I would strongly encourage you to make sure you have a realistic idea of what you are signing up for. The comments above reflect a lot of experience that should not be disregarded. Information you are putting together from family medicine programs that are trying to recruit you is going to be quite biased.
I hope that my situation will serve as a cautionary tale. Though given your response to the above comments, I'm sure you will find ample reasons to disregard my advice as well