You've got that completely backwards. Correctly applying Bayes' theorem, if an anomaly is observed you incorporate the prior into the calculation of the posterior probability. You don't just give up and say "the prior is miniscule so the likelihood is useless".

And then, even that's not enough. Decision theory needs to be applied to decide what action to take. That means taking into account the expected QALYs, cost and inconvenience across the distribution of possible outcomes. There's a whole spectrum of possible decisions, from immediately performing surgery, performing an invasive test like a biopsy, performing other less invasive tests, scheduling a follow-up non-invasive test at a later date, or just following a regular schedule of non-invasive tests and looking for any evolution along a longer time period.

The real problem is the binary thinking of either "we think you have X" and therefore tests must be performed or "we think you don't have X" and therefore tests shouldn't be performed. If medical organizations adopted empirically grounded decision frameworks, by applying them consistently doctors would be able to see something anomalous, assess that the risk isn't high enough to warrant immediate investigation, and be protected from a lawsuit in the unlikely case it was, in fact, something. And then we could do away with this "if we look we might find something" nonsense, which is pure fallacy.