A Sports Rehabilitator does not just do exercises with a patient. They have the clinical skills to complete a full subjective and objective examination from which to make a diagnosis. They can then treat with manual therapy, electrotherapy, massage therapy along with following that up with exercises to strengthen the injured area to reduce re-injury risk.
The anatomical knowledge is as good as a physio's, the whole first year is taken up by anatomy, physiology of injury, types of injuries (including mechanism of injury, signs and symptoms) and basic treatment techniques. All this is built on in the subsequent years.
I actually think that a sports rehabilitators choice of exercises is very specific to the patient. It is about tailor the exercises to the patients activities/lifestyle and being as imaginative as possible. I wouldn't give any exercise that I wasn't able to reason its use - I think that comment is unfair.
I believe that sports rehabilitators get bad press and that can be down to a lack of understanding from the other therapy professions and patients about what we can do. I have been given a great chance to work with a highly respected physio who recognises my talents, and to him it doesn't matter that I am not a physio, just that I am good at my job and get people better. Surely that is all that should matter, not the letters after your name.