On at least two occasions China has swab-tested 10 million people in a single city within a week, when covid re-occurred in a city where it had been thought to be extirpated. I don't think blood testing is twice as difficult as swab testing. So, I don't think it's impractical per se. It's just impractical in the UK.
You know what's impractical? 2.7 million people dying and an unknown fraction of the survivors suffering disabilities and injuries such as diabetes, heart attacks, and cerebrovascular attacks.