Humans got to where we are mostly because we're omnivores.

Simple enough. Why did humans become the dominant species on Earth, rather than, say, gorillas? Before Homo Erectus, what put us on that first step in the path to sapience?

Being able to eat any goddamn thing we want, that's what.

Not a bad starting point. This could have merit.

This would entail that goats are superior.

^ As well as tiger sharks and certain kinds of bacteria that literally subsist off of anything.

Neither goats not tiger sharks are true omnivores. Goats don't generally eat meat (they may eat something that has traces of meat on it), and tiger sharks don't eat plants (although they may eat something that has plant material clinging to it). Pigs, on the other hand...

Perhaps a combination of our omnivorous tendencies and our big ol' brains?

^ Actually it was those and our ability to distance run seemingly forever. We're the only species on Earth that can run for 30+ miles in a single day every day in all climates and terrain types.

In the age of technology however, most people can't run that far.

Well, gorillas are omnivores too. I think it's the tendency to pass learned skills to other individuals that eventually formed cultures, something where humans have progressed infinitely further than any other organism on the planet.

I've never heard of a gorilla forming a hunting party and gunning down tasty jungle mammals.

You're confusing "omnivore" with "hunter".

The fact that gorillas are capable of digesting meat doesn't mean they get the chance often at all.