The key to making an omelette, as I understand it, is high heat and lots of butter - more butter than you think you’ll need. As you mention, cutting it with olive oil is a good way to prevent scorching.
I like to whisk my eggs (three or four depending on how hungry I am) with a pinch of pepper and kosher salt, about a tablespoon of heavy cream, and a good-sized pinch of cheddar cheese. If I’m doing a plain omelette, I’ll pour that into the pan, occasionally lifting the sides up to let the liquid run underneath what’s already cooked, and roll it out of the pan when it’s all done.
If I’m really hungry, I’ll use that method as the basis for what I refer to as the MOAB (Mother Of All Breakfasts), in which I incorporate fried potatoes into the omelette, lay it out flat after it’s done cooking, and fold it around a mess of bacon, sausage, steak-ums, grilled onions and peppers, bleu cheese, sour cream, and hot sauce.
My favorite is simple: Beat an egg with added sriacha and spices. Pour into pan (I have a special one for this size) and misx for five seconds as you shake the pan. Let cook for a minute or two. place a slice of emmethaler cheese on the egg… When fully cooked, fold up 1/3rd of the egg. Then flip the egg over itself so it it’s a tube with the egg in the middle. Delicious!