This youtuber seems to have captured the essence of an omelette by trying, almost in vain, to replicate the technique of the famous Jacques Pepin... https://www.youtube.com/watch?v=5__zptEU9vE
There are certain spices that go well with egg, like tarragon, chives, shallots. Dried version of these work especially nicely with egg. You're using sea salt flakes right?
I like to froth milk with steam on my espresso machine and fold that into the eggs, right in the pan. The foam makes a nice creamy almost puffy texture.