I would put it this way: No producer can turn a badly composed, irrelevant song into something good.
I think opposite. Production is super important. It makes good songs SHINE, and lifts up average songs a decent amount. Imo, perfect example of this is
Magnum album
On a Storyteller's Night. Production is very good on this one, and even average or too poppy songs are enough pleasant to listen.
There's dramatic and more complicated - very good - songs:
How Far Jerusalem, On a Storyteller's Night, Les Morts Dansants
And more straightforward, simpler, not top tier songs: Just like an Arrow, Two Hearts, Steal Your Heart and so on.
Another perfect example is
Powerslave. Those two albums, overall, are very pleasant to listen. Production qualities helps to perceive those as ONE whole, and even average songs are acceptable, at least for me.
Good production makes good/great songs COMPLETE/FINISHED in every possible way.