It's way more than that. They are just more fluent at creating things like melodies and harmonies that "work" without essentially starting from scratch on every composition.

I know music notation and can write things down, but I'm not a good composer. I'm at the mercy of others to write new material if that's what I'm interested in playing.

And writing it down in a way that other musicians can understand is of nontrivial value if you want people to play your stuff.

Yes, being able to write is a good thing.

You can get so far with just an aural tradition but the invention of writing helps.