No, you have still misunderstood. The NHS model collects more data than the google/apple one. The idea was that that additional data would allow limited testing

No, you have still misunderstood. The NHS model collects more data than the google/apple one. The idea was that that additional data would allow limited testing capacity to be used more efficiently. For example, false positives for outbreaks would easier to detect, as well as wider trends. Also, it's not just tests that are short, but contact tracing resources. Basically, when you shut down a business or location due to an outbreak, you want to be as precise and accurate as possible - to do otherwise will cost the affected too much, and they'll start to ignore the contact tracing system entirely. We'll probably never know now if there was an actual benefit to the NHS's proposal, but it seemed at least plausible.