Bayes' theorem is not enough, I think you have to weight how expensive is the test, what are the consequences of false positives and true positives.

For example a low price test with almost zero both false positive and negative rate whose associate treatment has low cost and risk and that can save life of patients seems to me a test that should be applied to anyone at risk.

Bayes incorporates the false positive rate. In fact, using the posterior is exactly the point. It's just noting that the diagnostic power of many tests is not very high if the prior probability of the disease is low. All the points you make don't change that.

From a practical point of view, if the prior probability of the disease is low but the disease is mortal and there is a $0.1 test and $0.2 pill that cure the disease without having negative effects, I think that you should take the test and if positive take the pill. Knowing the posterior probability is not enough to decide what to do, you have to know what are the tools and trade-offs.

Wouldn't you just give everyone the pill in this scenario?

This is how non-prescription supplements work.