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And only the one person returning from South Africa would have needed to actually be a false negative. The rest may have been true negatives.

I know someone who just got a positive PCR test and still got two negatives on at-home test kits.

Relatively high yes, but with emphasis on the relatively; I would be surprised that there would have been more than 3 false negatives.

Note that one of the issues with rapid tests is that they only work reliably once you become symptomatic - however, you may already be infectious about two days before that...

I think you mean false positives? Or are they just bad all-round?

No, false negatives: you're tested as negative but you, in fact, have the virus. This is common for rapid antigen tests, and especially so if self administered.

False positives are not tipically a problem for this kind of tests. It's like the pregnancy test: if you see the stripe, you're very, very likely pregnant. If you don't see it, you can't rule it out with the same certainty.