Of course, the liberal revolution in France took the same step.
They had trials, but it really just came down to, 'The king is guilty of having been king.' The royal family is an existential threat to the new government.
In both instances, there was a civil war with one side having the royal family arrested, and the other side fighting to free them and reestablish the monarchy. Eliminating the royal family imply there is no going back possible, which must erode significantly the will to fight the civil war, no?