Firefox with XUL was this kind of browser.
Mostly because I'm selfish and want more of Org Mode's features supported in more places, but also I'm super impressed by Emacs' scripting/control abilities.
I'd also like to see vim with a mode specifically for Org Mode. But that sort of idea is very much beyond my talents :/
Anyway, my dream is a browser, a real one for modern web, built-in in Emacs. eww is nice but substantially useless and anytime I use Firefox (witch is well... Nearly any day) I complaint due to it's limited "window/buffer" (tabs) management, inconsistent keybindings support despite saka key, limited text navigation capabilities etc...
Imaging your (big) bookmark collection in an org file, ivy-searchable on demand, seamless integration for instance when you receive complex html mails (sgrunt), or for html feeds (sgrunt again) etc. instead of asphyxiated bookmark bar/manager + tab manager... Webpages as Emacs buffers, arranged as any Emacs windows, with consistent keybindings etc...
That, and pacing myself during mentally intensive tasks. I do 25 on, 10 off (more than pomodoro, because my brain). This works really well for me. You may have to adjust the variables to make it work well for you.
https://www.gnu.org/software/emacs/manual/html_mono/eww.html
The modern browser engine exposes a lot of controls to scripts and to the chrome layer. IIRC Firefox uses this extensively to code (most of) the UI in Javascript and HTML/ CSS. But this is not very exposed to the user, and web extensions are intentionally limited.
It appears that Next goes in this direction. I don't know if gtk-webkit allows to script it very deeply, e.g. enough to expose DOM and write extensions in Lisp.
It would also be great to have something like the Emacs's customization framework, and something like Melpa. (But this is obviously far down the road.)