I've only ever had SSSS to US, even though I believe it can happen both ways. The thoroughness of the search depends on where you are. Once you get to US no difference, except you probably won't be able to get precheck back on connecting BP. Yes, the suppression of digital BP is related to SSSS. Although my memory is once you board the flight with the SSSS (i.e. once that coupon is lifted), you could get it by redoing check-in. Have you tried that?

Getting it once is nothing to fuss about, just annoying. Usually your next trip will be fine. If you get it, say three times in a row, then perhaps time to write in for redress.