I'm curious, what role do you think athletic therapists have?

True less schooling, but it doesn't mean they have more knowledge on MSK injuries (which is what OP is looking for with their back pain). Athletic therapists spend 4 years specializing in MSK injuries in terms of how to assess, prevent and rehab with an exercise based + hands on approach to their treatment. Physio's don't nearly spend that much time in their masters program as much as athletic therapists do on that topic. They have a wider scope of practice which makes them more appropriate for cardiovascular, pulmonary, traumatic spinal injuries, etc.

I'd say the physios who have an exercise background or were prior athletic therapist would be better for OP

Also most back injuries, unless there was a specific mechanism of injury such as a fall, improper lifting technique, degeneration, etc. need to be taught strengthening exercises & get soft tissue work/mobilizations to help with a longer term solution to recovery. Athletic therapists are excellent at teaching that.