I can interpret your original query two ways:
1. Why can't we replicate Emacs like power in the browser?
2. Why can't we replicate Emacs like power in the browser using existing web sites as apps?
I was answering 2 above. If your mail application is Gmail, then non Google tabs can't read your emails and vice versa.
If you meant it from the perspective of 1, then you can get much further. You'd have to write a generic mail app in JS that can connect via IMAP, etc.
The other headache you'll have to deal with is not having easy access to the disk. Can you easily implement a file manager in JS and run it from the browser? Or access the processes on your computer (e.g. say you want to write a "top" tool for the browser)?
Essentially, the Emacs ecosystem has lots and lots of useful Elisp packages written over the decades. The browser doesn't. And the ones that exist are self contained (i.e. you don't find a JS TODO app that is meant to integrate with someone else's JS Mail app).
And browsers don't have an easy way to install a JS app. How would you do it? Bookmarklet?
As for customizing the browser via JS - yes, you're technically correct. The difference is that much of Emacs is designed to be modifiable via Elisp, whereas that's not the case in most browsers.
So to answer your question, it is technically feasible. You just need a better browser and a lot more people writing client side JS apps.