Next step: Browser becomes full screen and always-on. People don't know difference between OS and browser. Rich websites become icons on your new tab screen. People call them apps. Tabs become known as virtual desktops or spaces. Somebody invents an app that can show websites and calls it a browser.
While browsers relay on modern OS to function the reality is that you rarely interact with the OS from user perspective - only when you want to upload a file and such.
This is why the browser as a medium makes perfect sense for OS and I think this is why ChromeOS makes a lot of sense although we see it is not massively adopted - maybe not yet.
But yes, everything you just said is like strange dejavu.
The cycle repeats.