I had all this with my father during the first lockdown and after. Going to get a paper was largely an excuse to get in the car and get out of the house. Understandable but unwise at his great age. It is really all about clinging to independence for as long as possible and driving is independence. Nobody wants to give up independence but with dementia the person seems blind to the other side of the coin, that driving poses a risk to others as well as themselves.
My father continues to be delusional about his diagnosis and about driving. He keeps saying he needs to get another car despite his driving license having been revoked. ( I sold his car against his wishes, another long story). He has blanked out all the facts that he doesn't like, the diagnosis, the revokation of his licence, the refusal of the DVLA to reconsider, the two consultations with his GP who declined to say that of course he was fit to drive and that the diagnosis was wrong. In his mind he is a great driver and there is little or nothing wrong with his brain.
My father also has been difficult about social distancing and doesn't wear masks properly.
It looks very much as if this is something that will not go away. What will happen now iis that the DVLA will ask your GP for a report and I am pretty sure they will rely heavily on what the GP says. Now might be the time to ensure that the GP has all the facts. His licence might be revoked or a time limit might be applied to it.
I would worry more about the driving than the newsagent visits, catching covid in a very swift transaction like buying a newspaper isn't really at all likely but diminished driving ability has far more serious implications.