Part of the reason it took so long to roll out these lateral flow tests is that it was hard to find ones with an acceptably low false positive rate - it needed

Part of the reason it took so long to roll out these lateral flow tests is that it was hard to find ones with an acceptably low false positive rate - it needed to be well below 1% for exactly the reason you outline. (Apparently PHE estimates it at less than 1 in 1000: https://publichealthmatters.blog.gov.uk/2021/03/30/covid-19-reintroducing-confirmatory-pcr-testing/ https://publichealthmatters.blog.gov.uk/2021/03/30/covid-19-... ) Even when the rate of Covid infections in the UK was much lower than it currently is, retesting of people using PCR suggested that the large majority of positives were in fact true positives, at least amongst the adults retested. It's possible that the story is different for kids, but on the other hand they're not vaccinated in the UK whereas many adults are...