I think you are not thinking laterally enough. Think about arches, stones and the bible. See if that gets you any further

Those were in fact my first thoughts, at least keystone was.
It's obvious, I think, that the word key in all kinds of phrases indicates that a particular object is of prime importance. But his question was about the "birthday" of the phrase. I couldn't find any ref. at all to key player,
so I thought to try looking back to see when the first literal keys were invented, as their importance/necessity would have been immediately obvious.

As I said, it was the best I could do. I imagine that the
phrase key player was coined in the 20th century, probably in relation to some sport, but don't really know. It is also commonly used re: one person in a business deal.