Basically with the right education we are able to do what any doctor can do. We are 'specialists' is movement and pain. If we can assist to reduce pain to increase movement, why not add it to our toolbox.
I would not go straight for the meds though with most patients. I currently give them advice by telling them to go to the pharmacy down the road and telling them what they will likely receive from the pharmacist. If they need stronger meds, I refer to a doctor.

It seems to be a expensive route for patients to have to see multiple practitioners for one problem.
If they have a complex drug history and multiple comorbidities I would likely to not be happy to prescribe even if trained.

Currently "Pain specialists" are medical doctors who prescribe medication and monitor. They refer to physio occasionally (more often would be better), and sometimes prescribe exercise (often inappropriately).
These pain specialists are not qualified to prescribe exercises in a graded manner as physios are, so why not have a two way street. Better yet, insist on further training for doctors who would like to prescribe exercise and training for physios who would like to prescribe.