This is why you have multiple successive panels. If there's a disease that happens for 1 in 10k people, and you have a test with 1% FPR, 99 of 100 people will be false positive.

But what you can do then, is either run a more expensive, elaborate test or one that's proven to be statistically independent on the positive testing population.

FPR can even be a good thing. Let's say you have an expensive test with a very low false positive rate. Then you can mix together 100 samples, and get a test with a much worse FPR 100 times cheaper. Then you can repeat the same individually on the positive population.

This is fully automatic and you don't even think about it. Btw, this is why mass testing, and public healthcare can be better. You can amortize the cost of things across a large number of people for no disadvantage.