I've watched many line cooks at resorts-no non stick pans. Low to medium heat, melted btter or butter /olive oil mix in a small ladle into pan swirl, beaten eggs into pan, swirl, let firm up, add toppings, fold plate and garnish.
Well, there's a good chance that the reason I use non-stick only for my omelettes is because I really can't stand even slightly undercooked eggs.
When I make omelettes in my non-stick pan, I pour the beaten eggs in & immediately cover the pan with some aluminum foil. In about 2 minutes the omelette is cooked "well" without being overdone, & I can just slide it out.
Cooking omeltettes "well" in a stainless steel pan has never worked for me.