Note that solving the problem doesn't automatically mean knowing what the problem is or even knowing how to solve it

Of course. On the other hand similar things go for doctors. Just because a clinical trial has shown that given certain patients a certain drug helps those patients, you don't know why it helps them.

I plausible that some psychopharmaca work by reducing inflammation in the gut.

In addition most doctors who diagnose a problem as a misplaced vertebrae don't look at why the vertebrae got misplaced in the first place. A misplaced vertebrae is visible on an X-ray. The underlying problem isn't.

If you've leg pain and you visit both a competent orthopedic surgeon and a physiatric doctor and they can't help you, good luck with your leg pain, generally.

If pain is your only problem and you score decently on hypnotic suggestibility a good hypnotist can remove it. In general that's often no good solution because pain is a signal and you would want a fix for the underlying problem but simply removing pain can improving someone's well being.

The human mind is capable of simply shutting of a pain signal.

That's obvious, but there's significant overlap if we're talking about problems with the body, which is also obvious.

Doctors are essentially trained to be blind and not trained to perceive what goes on inside a body. That has advantages because the can give you treatments that work in blinded trials but it also has it's issues.

My own primary physician can't distinguish whether I tense up or relax in a conversation and she's a Yoga teacher on the side, so probably more kinesthetic than average.

The night I stopped being a bioinformatic student I thought "beliefs have to pay rent" as far as my nonstandard beliefs go. I went and fixed one of those problems doctors have told me I have to live for the rest of my life because I didn't feel as bound anymore to the scientific way of doing things.