JS has more features than it deserves for learning about and (critically) sharing information about the host platform. Yes, you can still learn some things as a website operator by watching what browsers load/don't load, and what they put in their requests.
Edit: I should hasten to add that there are other concerns beyond privacy, like accessibility and the fact that a web page has no bloody business deciding that I'm likely running an iPad and therefor I shouldn't have access to X or Y. This is dumb, and contrary to the idea of the open internet. It's the same thing that's wrong with this EME nonsense.
I can get behind most of what you say - as long as we are talking about simple, presentation based websites.
Where I think there's a breakdown in this view is when you consider complex web applications, including games. At that point, I believe some level of inspection capabilities are required, if we desire to have complex web apps delivered through the internet. I'm by no means sold that on-demand web delivered code is necessarily a good thing though. There's far too large a surface area to adequately secure while still making it useful, IMHO.