Only one out of 21 obstetricians could estimate the probability that an unborn child had Down syndrome given a positive test

Say the doctor knows false positive/negative rates of the test, and also the overall probability of Down syndrome, but doesn't know how to combine these into the probability of Down syndrome given a positive test result.

Okay, so to the extent that it's possible, why doesn't someone just tell them the results of the Bayesian updating in advance? I assume a doctor is told the false positive and negative rates of a test. But what matters to the doctor is the probability that the patient has the disorder. So instead of telling a doctor, "Here is the probability that a patient with Down syndrome will have a negative test result," why not just directly say, "When the test is positive, here is the probability of the patient actually having Down syndrome. When the test is negative, here is the probability that the patient has Down syndrome."

Bayes theorem is a general tool that would let doctors manipulate the information they're given into the probabilities that they care about. But am I crazy to think that we could circumvent much of their need for Bayes theorem by simply giving them different (not necessarily much more) information?

There are counterpoints to consider. But it seems to me that many examples of Bayesian failure in medicine are analogously simple to the above, and could be as simply fixed. The statistical illiteracy of doctors can be offset so long as there are statistically literate people upstream.