...wouldn't this show evidence that the stake was of pagan origin as well?
It sure would. Watchtower has stated that Jesus was actually murdered because of unfair violations of procedure at his trial. If the cross would not have been used to execute Jesus because it is pagan, as Watchtower says, are we to believe that the torture stake is a Christian instrument of murder? There is no such thing.
Stakes and crosses were both used by pagan societies in the first century. One is no more Christian than the other.