your vision is note impossible, but would be very, very difficult to achieve. hardly any physiatrists do exclusive sports medicine, and the ones that do are typically affiliated with an academic center. any specialty, be it FP, IM, ortho, etc. usually does mainly their primary specialty and a relatively small portion of their patients are sports patients. the general recommendation is to pick a specialty that you are the happiest with, and the sports can be added on as an adjuct if you are interested later.
that being said, it is not impossible. there are PM&R sports docs, and it is becoming easier to be recognized as such. the biggest problem is that primary care sports medicine is not really procedure-based, so you may have a difficult time getting paid as much as if you performed EMGs, axial injections, botowx, etc.
that being said, more power to you, and best of luck.
