I think part of it is that he is the President of the United States. Other politicians can go into self-quarantine, but that isn’t a realistic option for this or any president. If there is panic now, what will happen if the President is shown to be hiding in a bunker?
I understand that a reasonable response would be how bad would the panic be if the President got the virus and died, but I think that is where the line in the sand is drawn. Someone has to show leadership (and it doesn’t matter whether anyone thinks it is poor leadership) and the President must be that person, even at the risk to his own health.