https://en.wikipedia.org/wiki/User_agent
What you describe came and went, subsumed under the homogenizing effect of silos like Facebook. Mashups were the big new thing. There have been various attempts to bring it back, but the inertia of silos makes it hard to find anyone interested.
Though making the web this big publicly editable pile of documents would create a lot of spam. You already have to filter a lot if you have a form on a website, imagine if instead of filtering structured data, you had to filter diffs.
I think maybe a good half measure is just bringing web authoring tools BACK to web browsers! Netscape Navigator had one if I remember correctly. This would just allow you to write HTML files to disk though.
Bonus point for some standard protocol where you login with HTTP Basic Auth, and POST/PUT/PATCH/DELETE HTML directly to a page. You'd need some special server that understands that, but ideally it would be open for anyone to implement. You point your browser to https://yoursite.com/my/page.html and assuming you've logged in with HTTP Basic Auth, your browser suggests creating the page, rather than rendering the responded error page.
Edit: Protocol is the wrong word, someone correct me what this is really called. HTTP based... editing? A standard API? It's now something I want to make so I'll need to figure that out haha.
Edit 2: CamperBob2 in a sibling comment mentioned wikis. Just realized I'm describing a wiki lol
There was an effort many years ago to provide browser extensions that allowed users to mark up, edit, and annotate existing web pages in a way that would be visible to the community of users. It never went anywhere, unfortunately, probably because it would have gored way too many sacred oxen. (And/or it would have been co-opted by spammers, turning it into a social networking/reputation management project.)
And of course any such project that became popular would either ruin the creators as they try to keep up with moderation, or devolve into an unusable mess if they don't moderate.
Nowadays it's more important than ever to curate content well.
The job of the browser is to _render_. That's agnostic on whether you're creating content or just reading it.
Doesn't produce particularly modern HTML, but if you want to author HTML like it's 1999, it's out there.
In a parallel universe, web browsers are called webitors, and they can edit websites as well as view them. People can suggest changes. People can publish annotations. Web hosting is like email--pick (or build) any service you like and pick (or build) any client you like. The protocols will sort it out.