All those functions can be performed without the manager needing to be higher up in the organisational hierarchy.

And at places like Google, this is how it does work (on paper. In practice there are always exceptions, either due to actual hierarchy or individual charisma). If you look at Google's org, you essentially have 4 ladders: 1) Engineering, 2) product management, 3) business, 4) G&A superstructure. You can bet that if you're a PM trying to decide roadmap priorities, you're going to be driven by input from engineers primarily, not customers, sales reps or professional managers.

Certainly, and in some of the places I have worked with a good manager, they were not necessarily considered "higher up". These companies had separate technical vs manager tracks in some form or another. So the notion of "higher up" depended on the track you are on.