That would explain what actually happened better, but who would have had a chance to strike him? He was never planning to enter the battle until it was all but won. The plan was to have him think that victory is his so he can come kill Bran, and then kill him with dragon fire. There was no plan to hit him with that other stuff because it was much more difficult to do, and fire kills other wights.
Let John Snow have his big show down with him and ultimately fail despite landing blows with his Valyrian sword, and then let the Iron born hit him with a volley of dragon glass arrow heads.