Well, you're right. But honestly producers have been the most important players in the game since Rick Rubin when producers started to contribute more to songwriting (actually that's sort of a lie - I'd argue it started with Keith Olsen). It's sorta shocking that people don't know. I only even mentioned ghostwriting because some other comments mentioned it and because it actually does happen where the credits aren't in the open.
How Motown slipped my mind completely I don't understand but yes you're right, but I wasn't under the impression that at Tin Pan Alley the songwriters did the production themselves. That said, I hadn't done much research into that era of recording.
There wasn't really production back then, as that stuff was usually cut live to two track, then the vocals overdubbed. But the song writers were also the arrangers in many cases.
I'll add parenthesis to make it more clear.