As a surgeon and future intensivist, I agree that my cup of tea will probably not be a medical ICU
But at lots of hospitals, ICU's are not separated... for example several community hospitals.
I really believe the anesthesia, EM, and surgeons are well-equiped to take care of the multiple patient care types (neuro, medical, surgical, trauma) since, we are the most interactions directly caring for these patients. This is particularly true for the anesthesiologists. Add on that a variety of actual procedural skills, and it's clear that the medical trained people have a lot to learn.
At my hospital, a tertiary care large hospital in New York City, the Medical Intensivists rely the most on consultants compared to the other closed icus in the hospital. At the sicu the intensivists do their own dialysis, their own intubations, their own chest tubes, and their own tracheostomies, to name a few. The MICU does NONE of these (they will, occasionaly, but not routinely, do intubations).
WHat medicine people don't realize is that their training is inherently short. 3 years is a total residency for them. Surgery is 5. Anesthesia and EM are four (granted, there are a few 3 year EM programs, but those are fading). That extra year makes a huge difference in volume/experience.
As a surgeon, I have not had the outpatient medical experience taking care of patients like my medicine colleagues, but I certainly have taken care of just as many,if not more, critically ill patients with a variety of co-morbid medical conditions in the context of their surgical illness.
Ask any ICU nurse generally who are their favorite residents or fellows or attendings in the ICU. In any ICU they generally love the surgeons.