The positivity rate is lower than you might expect from the asymptomatic testing in Liverpool, especially given what people are pointing out that those who work outside the home or have mild symptoms (headache, sore throat etc) being more likely to get a test.
One possible reason might be the type of test they're using. The lab processed tests pick up a number of people as positive who've had Covid weeks ago, many of whom will be asymptomatic at the time of the year. These new instant ones work a different way and are intended to identify those with current, active infections. So you'd expect fewer asymptomatic people to have sufficient levels of infection to show up as positive.
Another possible explanation is that Covid has been so much in the news in Liverpool that many people have been getting NHS tests when they have any symptoms at all, or possibly even if they don't, but have had close contact with a case. Unlike the asymptomatic people who are picked up in the ONS survey many of whom may have no idea they're at risk.