I'm an MS-1, so I'm just passing on what I've been told, but we had a panel of ob/gyns talk about their practices today and they seemed to be in agreement that the general ob/gyns are the "point of entry" for many women into the healthcare system, meaning they give flu shots, take BPs and check blood glucose but they don't necessarily manage respiratory illnesses, hypertension or diabetes. If they see a patient with a non-gyn problem which needs treatment, they might treat the patient themselves or, perhaps more commonly, they might refer the patient to an internist or a specialist. It sounded like given the fact that they see many of their patients yearly, they were frequently exposed to general primary care issues but that they didn't necessarily manage all such conditions themselves.
It's also worth noting that they pointed out the various opportunities for subspecialization within ob/gyn (gyn-onc, maternal-fetal medicine, reproductive endocrinology, etc), so if you don't like primary care, it seems like going down that road would provide you with some options, too.