You can certainly check your zero by firing a single shot, provided you are happy that you squeezed off the shot correctly, but when setting the rifle up initially it is necessary to fire groups of three so as to confirm the centre of the group.
You might make coarse adjustment after a single shot but the fine-tuning needs a group.
A close group confirms that the rifle ‘likes’ the ammo, that the first shot was not just a fluke, and gives the shooter the comforting knowledge that if a second or third shot is required the bullets may be relied upon to land in the same spot.
An accurate shot needs to be repeatable and the group tells us whether that is so.
However, as you suggest, once the rifle is zeroed, a single shot is usually enough to check that nothing has moved.