At our hospital, we do. Some holistic practitioners even have hospital privileges to come to the hospital, at patient request, to perform acupuncture treatment.
From my experience, the best approach is to encourage BOTH if the patient wishes it and not make it an adversarial choice of "one or the other".
If a patient wants to take herbal pills that may or may not be proved effective, in addition to proved therapy, so what? If it makes the patient feel more confident, that's great.
Where I have a huge issue is where some "natural" healers push their agenda as an "either us or them" with tragic consequences that I have witnessed first hand. One "organic" patient that had a curable cancer that was melting away to the size of a grape suddenly "stopped the poison" of proved chemotherapy to go off to California to have some "curative herbs" administered only to come back nine months later with a recurring tumor that was, literally, the size of a football.
Resumption of proved surgeries and chemotherapy saved the life but that patient is now horribly disfigured. It is much easier to finish killing off a small grape than it is to kill off a football that has eaten away some rather important bones in your body.
Controlled experiments are not Rocket Science. Any 8th Grader who has studied the Scientific Method knows the basic principles.
Even if somebody comes up with a once upon a time seemingly ridiculous hypothesis ("Peptic ulcers can be caused by a bacteria named H. pylori"), once that hypothesis is proven with controlled studies, the World will beat a path to his door.
So, in your case:
You take large enough series of patients with the same condition that you had. You divided them into three groups. With one group, you do nothing. With another group, you administer sugar pills and "treatments" that you invent out of thin air. With the third group, you administer the treatments that you had.
You wait however many months your treatment is supposed to take to work. You test the patients for improvement.
If the groups show no difference in results, your result was probably a fluke related to your own body.
If the group that used your treatment showed statistically significant better results than the other groups, then your treatment gets published as a proved therapy and joins the ranks of other treatments in evidence-based medicine.
Six of one, half a dozen of the other. I worked for a natural foods store that included medications. There is much that is complete and utter bunk, but there is some good stuff in there.
The key, as always is to trust, but verify.