In order to maintain confidence in vaccines in general this sort of thing has to be taken seriously. You can't say that the vaccines are perfectly safe and then have a bunch of anti-vaxxers shout that people are dying from blood-clots and no action is being taken. That would severely damage trust in the health authorities which would lead to fewer people getting the vaccine from any company, not just AZ.

Better to stop everything, be clear that action is being taken, transparently evaluate the risk and then start back up or stop as appropriate.