R & B: Smokey Robinson: Holland, Dozier and Holland. Metal: Haven't followed that genre too closely but Kurt Cobain would probably qualify. Classical for the most part was music but very seldom any lyrics which is probably why the likes of Bach, Beethoven and Mozart wouldn't fit in here. I mentioned many folk oriented songwriters including what I consider the Big Three: Dylan, Lightfoot, Kristofferson. Oh yes I guess we need to mention John Denver also.
Where 1950s and 60s country is concerned, no one yet has mentioned Buck Owens and Roger Miller.

I recall one personal asking about the earlier days of pop music. There were very few so-called celebrity songwriters during that time, Irving Berlin probably being the exception. The singers and songwriters were seldom one and the same at that time.