Firstly, these consults are never as easy or logical in real life. If you think doctors don't understand Bayesian probabilities then how well do you think patients do?
Decision making when screen positive results occur, with appropriate perspective, is virtually impossible for patients. I've lost count of the procedures I've done for incidental findings found with screening, which we KNOW are benign and will remain benign. Unfortunately, despite very strong reassurance, patients generally want these incidentalomas removed.
Secondly, unless your positive screening test can be dealt with.... a) by a procedure that has zero risk of complications and b) in a way where no stress is engendered while waiting for definitive results and) the costs of (screening + treatment) could not be more effectively spent elsewhere ...then there will ALWAYS be negative sequelae with screening. Therefore these negatives need to be assessed in a real world RCT of sufficient size and with maximal control of bias, to make sure they don't outweigh the positives.
Your perfect world of logical patients balancing pre- and post-test probabilities, and acting accordingly doesn't exist.
IMO, if pre-med requirements included a bit more statistics and a bit less biology, we'd all be better off...
"last month, the American Cancer Society urged more caution in using the test"
It's a very sad state of affairs when a test that unambiguously offers some statistical utility (speaking as a Bayesian, any test is useful if we do the right things with the results, this is quite literally a provable theorem of decision theory) is potentially problematic because we aren't properly evaluating what to do when we get the results. I can't blame patients here, they're not the experts; it's doctors (and perhaps more fairly, the entire medical research industry) that need to wise the fuck up and realize that they're acting as statistical dilettantes in a field where they're putting patients lives at risk because they don't understand math very well.
Yes, even a test that only detects ~3% of cancers should be useful; but this requires that doctors completely understand what it means both to see a positive and a negative result, and don't overreact when they see either one.
If this was a matter of just weighing the costs of having tests done versus the benefits of getting the results, it would be one thing. But it's not. People actually end up suffering and spending great amounts of money just because they did "the right thing" and had tests run, and then listened to the doctors' advice afterwards.
Once more with feeling: after discounting for the cost of the test, there is no test that should be of negative value for a rational agent to take. The results should always be used in a way that, in general, increases the welfare of the people taking the test.
If this is not the case in medicine, then they're doing something seriously wrong mathematically speaking, and this is a very bad thing that should be a high priority to fix.