Personal opinion - deep fried eggs are goated. It's either runny scrambled (pretty hard on cast iron/stainless) or deep fried. Anything in-between is incompetence.
Runny scrambled (French omelet) isn't too bad on carbon steel IME. And certainly possible on cast iron, since that was the type of pan used when Carême wrote his cookbook. By Escoffier's time carbon steel was more common. Neither of them is famous for having invented a time machine, both predate non-stick cookware, and they're the most famous French chefs in history.
French omelet is completely different thing. I do use my cast iron for this, it's okay if you go hard with silicone spatula. But still, more work than nonstick. I keep meaning to get carbon steel!