"It seems pretty intuitive, so maybe it's already largely applied?"

It is! That's why doctors don't order every single possible test every time you show up sick!

That's also why consumers SHOULD NOT be testing by themselves without medical supervision (I'm looking at you, Elisabeth Holmes and Theranos).

It is at least implicit in testing practices; the reason diagnosis often involves a series of tests is that they first screen with cheap, low-false-negative testing, then apply more expensive tests to weed out the false positives. But of course the rarer the disease, the harder that second phase becomes.

In one of my courses in medical school, our professor, a physician, told us that many doctors follow Bayesian inference without even realizing, based on their experience and education. The best doctors recognize this ang educate themselves on epidemiological studies to better use tests and procedures and medications to take care of patients most effectively.

I've actually been going through this recently with a series of cancer tests with a high false positive rate. When I was discussing it with my specialist he said that he and other doctors don't necessarily think of it in rigorously Bayesian terms, but they're familiar with the sensitivity and specificity of each test and have a pretty strong understanding of what that means based on priors.

Talking to medical students I get the same impression. The refrain I commonly hear is that high sensitivity tests can rule something out, and a high specificity test can rule it in. Obviously that's a reasonably shallow understanding, but when they combine that with their intuition of a prior it seems like they get pretty good results.

I don't think it's that they get good results, but moreso that it's the conventional wisdom (or "standard of care") and a good mechanism for learning from and properly attributing bad patient outcomes just isn't there. I doubt anybody has told them that their patient died or was given inappropriate treatments because they didn't have the proper appreciation for Bayes theorem, even if it were the root cause. That's just not the culture of medicine today.