Testing: of course they aren't testing everyone who feels a bit fluey.
There is only a finite number of test kits, and there's a whole load of other things with similar effects (common flu, colds, etc). If they tested everyone who came up with a sore throat, there would be queues miles long (putting people in close proximity, which they don't want), they wouldn't have staff numbers to deal with it (irritating and provoking complaint), and they would run out of kits/analysis capability (irritating those queueing, and preventing them testing those who really need it).
I don't think there is a 'win' scenario that they could use for everyone.
The government waffle are not helpful because they are non-specific, coupled with a distrust that politicians will tell the truth. The health officers are clearer and should be dealing with data. The politicians should take note of Klopp's response when asked.
We're a few days back from Italy, and also protected to some degree by geography. Being a bit behind, the need to take dramatic (and unpopular) action is less urgent, so they won't. I'm quite impressed by the Italian approach to dealing with it - quite tough and popularity be damned.