If you stuff something covered in egg yolk in there it's coming out with egg yolk on it. If you don't scrape whole chunks of potato off the plate, you're rolling the dice with whether you pull it out of a filter or out of the thin hose between the dishwasher and the drain. The dishwasher's not magic - if it takes a scrub to get it off, you need to scrub it.
I don't know about handwashing after it comes out though, that's crazy.
Egg yolk is no problem whatsoever for a modern dishwasher and detergent. And obviously you don't put whole potato chunks in the machine. Don't be stupid.
GE machines come with a garbage disposal built-in. Be as stupid as you want, things are still coming out clean.
https://www.geappliances.com/appliances/dishwashers-with-pir...