> IPC, memory management, process management, network management.
I imagine the non-trivial parts of these are done for the JS VM (correct me if I'm wrong), and therefore a VM design course would have more intersection with Browsers with respect to these disciplines than an OS course.
> security
This one is everywhere, it has no special connection to OSes.
> They all happen at a slightly higher layer
Slightly?! That's an understatement of the week! The difference in abstraction levels is huge here and the specifics of the two levels are very, very different.
> in terms of what you need to know, an OS class or three is definitely helpful
Sure. But I think it's as useful as any systems programming course. I can agree that systems programming is a good preliminary for both Browsers and OSes, and learning either of the two will teach you a good deal about systems programming, but I doubt they will repeat each other.