That was so that the members of the Firing Squad would feel no remorse or guilt. No one would know which was the blank, and each would then think that they could have had it.
If you stop to think about it, that would mean that someone else would have to load their weapons for them. If an individual on a Firing Squad was tasked with loading his own weapon, he would, as an experienced soldier, recognize a blank round. [Edit to add: If he had an empty rifle, he would easily recognize that when he fired.]
So following that further, perhaps there were no Blank rounds, as they would be difficult to hide or fabricate to look like real rounds, which would cause problems of another sort; and if you had poor shots, then you would be increasing your odds that the execution would not be 'clean'. Perhaps, they lied and said there was a Blank, when there was none, just to remove the guilt of any member of the Firing Squad who had doubts of what he was doing.
Even in Aid the Civil Power, if the Riot Control Box that was deployed reached the stage of using 'Deadly Force', there was another member whose task it was was to go out and pick up all the casings ejected by the 'Riflemen'. That way the 'Riflemen' would not have to.