Matt, It's not the chiffon cake that is affected, but the angelfood cakes that can fail to launch if a trace of grease remains in the pan. Apparently this was more of a problem when soap was used to clean dishes instead of modern detergents. Mom always removed the chalaza from the egg, too. It is the part that suspends the yolk in the lining membrane of the egg, one at each end. Nothing wrong with it, but I guess it looked icky to them.
Have you ever seen old electric skillets that had brown baked on grease on the outside? It's a result of them being washed last and lowered into the water when the soap or weak detergent was overwhelmed and grease was floating on the top of the water. It adhered to the outside of the pan which was not washed as well as the inside so the grease baked on. I guess with stuff like that happening in the dishpan, the warnings about keeping grease away from the angel food pans was spot on. I remember ladies with Revere Ware with baked or burnt on grease on the outside of the pans, usually with a gas stove too, or how else do you get burnt on grease near where the handles attached to the pan? These people would say that they did not think a dishwasher would get pans clean. I would listen to them and go home and look at our pans with not the first sign of burned on grease or any grease even before we got a dishwasher and just knew there was something wrong with the way they (did not) wash their pans.