I agree that using Eddies as troopers is far better (= easier) than non-Eddie toons and I usually have Eddie as a trooper for this reason, too.
However, there is one time I switch it to non-Eddie as an exception - when there is a challenge requiring skills of specific ultrarare character. That's when I change my trooper to such character (e.g. Musashi lately) - to help my troopers to do the missions they might not be able to do otherwise. This is the exactly same reason I am thankful to my troopers who list their Raven Witches or Charlottes (at the times when I also find myself on the other side of the stick).