I'm very late to the party with this post, but wanted to weigh in.
Why are the allied health professions not represented on Star Trek?
As a practicing physical therapist, I think these roles still are relevant in the world of the future. Here are my reasons why, from the point of view of my profession.
The first thing that everyone needs to keep in mind is that human mobility is way more complex than simple musculoskeletal injury. It's not just the musculoskeletal system, but also the central nervous system and the motor map in the brain, the endocrine system, the cardiovascular system, as well as the person's thoughts, beliefs and energetics around how they move that make movement happen. Therapy treats all of this!
With their current training, a general practitioner, or even most specialized MDs, can't do what a therapist does. In the hospital, if the person can't just pop up out of bed (and much of the time they can't), doctors don't know how to instruct them in the correct techniques. If the person's vital signs are unstable, doctors know what the safe range is for exercise, but they don't know the practical piece of how to actually do that during an exercise session. (I've seen them try and it never goes well. :) I also frequently got orders for patients that actually were not appropriate for exercise once I reviewed their lab profiles, and I spent a good amount of time educating the medical team on when the person could safely start activity. What I do is a completely different skill set than what they do.
This is also true in the outpatient orthopedic setting. When I get an order from a doctor, it contains a very generalized diagnosis, but it is up to me to figure out specifically what is going on and decide a plan of care. I've also seen that docs have no idea how to teach or progress an exercise program. They're great at what they do, but therapy is a different skill set.
In order for a general practitioner to be qualified to do what a therapist would do, they would need at minimum an additional three years of training beyond the current medical curriculum. This does not even account for the interspecies component that would be present in Starfleet! I think it would be highly unlikely that the doctors of the future would add this component on to their curriculum, especially since the therapy skill set is fairly different from what they do, and most GP s have enough on their plate without this.
For technology, it could take over a part, but not all, of what therapists do. People tend to think that pain and acute injury is a large part of what physical therapists do, but in reality it's just the tip of the iceberg. Sure, a medical device can definitely handle the simple acute stuff- someone broke their hand/nose/etc. on the holodeck after tripping over loose gravel and falling onto it. However, the majority of patients that need therapy are in for more complex stuff.
Example: A patient comes in complaining of knee pain while running. You could run a tricorder scan and see that the person's kneecap and IT band are inflamed. A medical device can definitely fix the acute inflammation. But...did you fix the symptom, or the cause?
You fixed the symptom, of course! The person will feel better for a couple of weeks and then come back to you with the exact same problems. This is because you never fixed the cause.
If this same patient goes to a physical therapist, they will do a gait analysis of the person while running. They could notice any number of irregularities in the way the person runs. Some of these have causes in the musculoskeletal system (stiff feet, flat feet, torsion at the femur, weakness of muscles, and the list goes on...), some of the irregularities are due to other systems, and many, many more of these irregularities have causes in the motor patterns of the brain. Even after the physical therapist and the doctor collaborate to address all of the musculoskeletal and systemic things going on, not just the initial painful symptoms, those motor patterns still need to be retrained, because they often don't automatically straighten themselves out once the musculoskeletal items get fixed. This is still the job of therapy.
There doesn't seem to be a technological thingie in the 24th century that you can point at someone that will automatically retrain their motor patterns. If there was, people would have no need to actually learn a sport. Captain Janeway would not keep getting knots in her neck and upper trapezius muscles. (Voyager: Resolutions) And Ensign Nog would not have phantom limb pain after he lost his leg (DS9: It's Only a Paper Moon).
This is an example from the PT perspective, and I believe that other allied health professionals could offer similar ideas from their own point of view. There was a good argument made that our roles would change, such as being less involved in the management of acute injuries, which makes a lot of sense. However, I think that the needs of the crew would still be such that having onboard allied health professionals would make more sense than having to ship a crew member back to a starbase (and thus losing that person's contribution to the ship) every time they need therapy. After all, most people pursue their rehab program while still living their daily work and social lives.
So, why do the allied health professions never show up in the series?
I think it's for the same reason we never show up in any medical TV series. What allied health practitioners do takes patience and skill and perseverance, but it's not sexy, and doesn't make for good TV. :) Instead, we work quietly behind the scenes to help get people better.