countries should be testing every single person every time they interact with someone
That would be nice but the UK's current test capacity (well, last time I saw it reported) was only 4000 tests per day.
With 560,000 frontline NHS workers it means it could test each of them every 140 days (assuming no-one else is tested).
To test everyone in the UK at that rate would take just under 46 years.
Whilst so few tests can be performed every day there is little point worrying about testing[1]. If the cases continue to grow at the rate they are it will only be a week or two before the number of new cases each day outstrips the current testing capacity.
I've no idea why the UK's testing capacity is so low compared to other countries.
Also, the current test (for the virus itself) takes 3-5 days to return a result. Someone who gets a negative result could be positive by the time they receive their result.
Someone who has had the virus (possibly entirely asymptomatically) could return a negative on the 'is the virus present' test. AIUI there is no high capacity antibody test to tell whether someone has had the virus or not and whether they display immunity (to the current strain[2]). It's also unknown for how long someone would still likely to be contagious even if they test positive on a suitable antibody test - they would need to perform both live virus and antibody tests together and there's still a delay for the results in which time that person is in limbo (not so good if they are a frontline NHS worker).
It really isn't as simple as "we just need to test more" (although that is obviously true).
1. That doesn't mean we shouldn't be throwing resources at it to ramp it up.
2. If it mutates significantly then all bets are off.
