Okay, so to the extent that it's possible, why doesn't someone just tell them the results of the Bayesian updating in advance?

Because then they would be assuming they had all relevant prior information for that particular patient. They don't.

For example, age of mother, age of father, their genes, when they've lived where, what chemicals they've been exposed to, etc., are many factors the manufacturer has no knowledge of, but the doctor might. Naturally, it would be helpful for the company to make an online diagnostic model of all known relevant factors available online, updated as new information comes in, but given the regulatory and legal climate (at least here in the US), something so sensible is likely completely infeasible.