"If you're good, you'll climb naturally." Is this statement true?

Eventually.

Honestly, there's a lot of factors involved, but you are the variable you can control. You'll eventually climb. If you're not good enough, get even better. Improve. Focus on what you can control.

Faker and other LCS players didn't get there through luck, after all.

Well, you haven't given any "To - From" but, in short:

Yes.

Of course not. Luck plays a decent role in climbing as well.

Not really, the game is filled with entitled 12 year olds who threaten to afk or feed if they don't get their way.

At this current point in time? No. But generally, yes. There's a lot of folks they need to ban for a few weeks.