// Can someone explain what he should have done? //
No problem.
Asked any number of friends, or the family he and his wife have nearby, or ask his government colleagues for some advice, or done what everyone else is asked to do - dial 911 and ask them what they think.
The government message was 'Stay home, protect the NHS, save lives'.
The idea is, avoid infection, or spreading infection, and as a subtext of common sense, stay off the roads and avoid a potential accident involving you and innocent people, putting NHS and police lives at risk while they deal with it, and stretching resources even further without good cause.
Isolate yourself or your wife in your large house.
Don't presume a worse case scenario and act in advance of it - deciding you need 'child care' and then it turns out you don't, you just need food shopping, which you could have got without leaving your home.
Don't 'test drive' your eyesight with your wife and child in the car.
Accept that as a high profile government advisor, breaking the rules will be discovered, and you and the government will be hung out to dry.
When you are found out - fess up and apologise.
Don't hide for a few days, and then come out and talk garbage about your house being 'under threat' when you drove back to it at the earliest opportunity.
Don't be an arrogant twit, you have done wrong, say so, say you are sorry, show some empathy for the hundreds of thousands of parents without the luxury of 'childcare' which you didn't even need.
Show some genuine humility and regret.
Think ahead next time to what you are doing, what the actual benefits are, and weigh them against what the consequences could very easily be.
Stop bleating.
Stay away from cameras and microphones for the foreseeable future.
How's that?