I’m still not sure what a project manager contributes.

I've well over a decade in software project management. The number one thing we contribute to a project is saying to the client (internal or external) "Sure, we can add that feature but it will have an impact on the delivery timeline unless we deprioritise other features. Are you happy for us to extend the deadline? If not, let's talk about what we can cut from the existing scope in favour of your new feature."