[hansvm]

Fresh eggs make it easier to get the right texture, especially for new cooks (less albumin separation -- same reason fresh egg whites whip better). If you're already past the basics, freshness won't impact the result much outside of, perhaps, a bit of weepage.

Egg flavor is fairly resilient though. Even repeatable effects from, e.g., flaxseed in the diet, are pretty small. You might notice them, but in a french omelette the egg is a smooth base for delivering the actual food -- butter. I'd experiment there (very easy to make your own cultured butter at home) before playing with the flavor of eggs, and personally I basically ignore flavor when evaluating which eggs to buy.

For anyone else reading along, room temp eggs also make it much easier to get the right french omelette effect, especially without a nonstick pan. 5-10min in warm water if you refrigerate your eggs should suffice.

[doublepg23]

I have heard fresh eggs making superior poached eggs for the same reason.

[hansvm]

They absolutely do. There are cheats and hacks you can use to make decent-looking poached eggs regardless (molds, vinegar and salt in the poaching water, double-cooking with an intermediate ice bath, straining off the weaker outer proteins before cooking, ...), but fresh eggs give a more consistently good result.