There are two separate tasks that need to be filled as a company starts go get beyond a small size. One is the coordination that you mention. The other is the day to day people management: making sure people are happy, understanding their hopes & desires, steering them towards growth, that sort of thing.

What I've found over time is the implementation of this is often very inefficient.

The coordination aspect is often a blend of the product organization and the engineering leadership, and in both cases I find there's too much authority pushed all the way down into individual scrum teams. There needs to be a cohesive product vision across all of the teams, and some set of people need to make sure they're all working on roughly the same things at roughly the same time. But that's about it. And you don't need a Product Manager, an Engineering Manager, and a Scrum Master for every set of 8-10 developers in order to make that happen.

The people management portion is trickier. You *do* want to cap that at a ratio of about 1:8-10. But this role is traditionally associated with authority over things like on what their reports are working. Which exacerbates the point above.