Not really sure I understand your second paragraph. You can have a high absolute number of false positives with a tiny FPR only if you have a very high volume of negative samples. This isn't an issue with looking at the FPR, it's an issue with not knowing what FPR is acceptable to you for your particular application.

The ROC curve does not assume anything about your positive:negative ratio; the PR curve does, so if the ratio in your test set is different from your ratio in production (and often you don't know what the "true" ratio is in production), your precision measurements will be misleading.

A general difficulty with very low FPR or FNR measurement is lack of samples to measure on e.g. if you have 10_000 negatives and your FPR is 0.1%, you're only estimating your FPR on 10 samples, so the estimate will have high variance - but I think this issue would affect precision and recall measurements at the extremes as well, right?