That's exactly the reason why neither I, nor anyone I drink with, is prepared to submit our contact details to the government when going out for a beer. That's our decision based on the fact none of us fancy being told what to do on a crude, non-targeted, blunderbuss basis without even proving that we actually have the virus and are carriers. If they want to give me a virus test, and then pay me my full wages for two weeks if I test positive and they want me to isolate, that's fair enough and I would have no issue with cooperating.
What they have implemented though, is a crude "you might have come into contact with someone who has the virus, so we want you to stay home for two weeks and not be able to earn a living or go about your business, and BTW you have to bear the cost" regime.
I'm not at all impressed how track & trace is being done. If someone in my local comes down with the virus, the logical thing would be to send out the testers to visit the pub and offer all the regulars an on the spot test - and guarantee that anyone testing positive would get 100% of their regular wages paid if asked to isolate. That would work as no-one would be any worse off by co-operating. The system they have got, you are personally better off staying anonymous and refusing to give the government any information, which undermines the effectiveness of the tracing scheme.