It's much, much more collaborative than that. The role of a producer can be limited to managing the financial side of the production. Or they can be involved with coordinating the production itself. The director has the most say over what is shot, how it's edited, what the score should be like. A good producer can help make a film excellent, but if you have a bad director, or the "wrong" director, then nothing the producer does will fix that issue.

Of course there are also Executive producers who are far far up the financial food chain of a movie, and sometimes these titles are also just vanity plates for the production.

As an example of the value of a good/great producer, look at Gary Kurtz. He was essential for both Star Wars and The Empire Strikes Back, but wasn't part of the team for Return of the Jedi. In my opinion, that's one of the reasons that ROTJ is the worst of the three films. Lucas didn't have someone pushing back against his worse tendencies.

For an example of the wrong director, look no further than Rogue One. While it's very difficult to sus out what was wrong with Gareth Edwards directing, Tony Gilroy was brought in to fix the film. Both are appropriately quiet about what the changes were, what was re-shot (simple professional courtesy), but the result is a film that (while I love it) is just slightly off.

You’re talking about something different. Certainly the director is the creative force. The sine qua non.. (Or.. cine qua non??)

But in the context of this discussion, the producer is the one that coordinates and brings together the huge complexity of the project. That is their defined role. It is the director’s vision and decision-making, but the producer ensures the execution across hundreds of people and many departments.

(Source: i worked in films)

"Or they can be involved with coordinating the production itself."