Have you any idea how complex a browser needs to be? I hope this team of plucky, idealistic coders can keep up with all the latest and greatest web developments devised by the thousands of engineers at Google et al.
The web as an open, human-sized system is dead.
I think a decent layout or CSS library would be useful outside of a web browser, too.
Then I would also only focus on the subset of websites that are "documents", not "apps". If I could decide, HTML6 would have two profiles: one ultra restricted (maybe no legacy stuff and no cross site scripting) for "documents", and one where you can do all kinds of crazy stuff like "web USB" for "apps". That's not going to happen because Google and Apple like the "open" web as complex and messy as it is, because it gives them total control as you know. But it doesn't stop a browser vendor from building a browser with two engines - your own engine for the majority of documents and chromium for webapps.
We tried to introduce office 365 company wide for collaboration, but we had a whole plant of 150 employees "mutiny" because they refused to use any of the office web apps because they were too slow.
This slowness is not a fundamental problem, but incidental. The layout calculations of even the most complex websites could be solved by a 10 years old computer instantly. The rendering can be done by a GPU without sweat. Look at how many polygons and shaders games had 10 years ago.
And note I'm not talking about "lightly formatted text files", but 90% of all web sites. News sites, Youtube, Reddit, .... What I'm excluding is Gmail, Office365, anything that uses Vibration, NFC, WebMidi, and other boutiqe features (you could probably add those back as libraries if you need them later). I'd argue that not supporting "apps" is a feature - you'd have to explicity allow them and they'd run in a sandboxed tab.
Something for regular people would be many more years of work.
Apple hates web apps. These are pain in their ass on the way to forcing every app to go through their app store with all the dystopian control it implements.