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I think it's all about personal preference and budget. I can make good arguments for the All-Clad stainless d5 non-stick "fry pans" I use for the purpose, I can turn out a decent omelette using a beat-up steel pan in a restaurant or a scratched-up aluminum relic in a rental.
There's a lot to be said about how a good tool feels in your hand and the joy that comes from using one instead of just "making do". The even heat transfer of a good pan makes is much easier to avoid browning the eggs (which is a sin in my book).
You might want to take a look at All-Clad's d5 omelette pan (which just happens to be on sale at Williams-Sonoma right now).
I love my 10" tin lined Mauviel copper pan for omelets. To me, it's perfect in every way and have been using it for 30 years.
I have only two non-stick pans and they are both exclusively for eggs. They are both from Calphalon -- thick enough for this application and a good value. Sooner or later, we all toss out a non-stick pan with regret.