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.
