> whether it is ethical or worthwhile to expose patients to the risk of harming their health through the follow up
So you seem to be saying that, in certain cases, if you have a positive test result, then the expected value of taking certain follow-up actions is negative. It follows that a rational actor, knowing this, would not take those actions in that situation. Then isn't the solution for doctors to update their procedures so that they don't take those follow-up actions in those situations (and explain the odds to patients who care)?
The phrasing "expose patients to the risk of harming their health through the follow up" makes it sound like the follow-up is something that automatically and unavoidably just happens, as if no one has any agency in the matter. If that's true, due to some kind of regulations or rules or liability rulings, then that sounds like a problem.
To some extent this is resolved by backpropagating: if the test is just a two-value "positive"/"negative" thing, and you plan to take the same action (i.e. nothing) regardless of the result, then there's no point in taking the test. However, I expect there are also other tests where, say, the test has a "super positive" value (or value range) where you should take action, and a "technically positive" value where you shouldn't act (plus a "negative" value); and the test is worth taking because of the likelihood of "super positive", but that means you do sometimes end up with the "technically positive" result, and must solve the problem of knowing when not to act.