When you "bookmark" some page, a snapshot of its contents should be saved to your disk.
You should be able to switch into "research mode" when every page you visit is saved to your disk.
Everything have "bookrmarked" should become fully searchable, as well as navigable in a variety of different ways.
If you "bookmark" two pages that link to one another on the Web, you should be able to navigate from one to the other locally by clicking that link.
When you bookmark a page an algorithm should determine which of the existing folders is shown as the default. Even when the default is wrong, you should never have to click through multiple interfaces or scroll through countless folders simply to save something.
There should be a way for you to instruct your browser to periodically revisit a "bookmarked" page and store its snapshots if there are significant changes.
You should be able to generate a cryptographic key pair right in your browser.
You should be able to use your private key to sign any piece of text you put into a form, regardless of the website.
You should be able to easily download public keys from people's profiles on various websites and subsequently use them for verifying signed piece of text.
You should be able to register your public key with a website and use your private key to log in.
It should be possible to embed a snippet of HTML or JavaScript into a page without giving that code ability to hijack the entire website. The fact that this is not possible is truly mind-boggling, since it means core Web protocols do not have any support for collaboration.
Let me remind you that web browsers have existed for over three decades. Let me also remind you that they are used by billions of people and are the primary tool for accessing everything on the Internet. None of the things listed above are particularly hard to imagine or implement. Their absence is highly conspicuous.
Note: I put the term "bookmark" in quotes, because I believe it's an absurd term for what we really need. A bookmark is something you put into a book that you have, not a reference to a book that you don't. A bookmark in your browser (right now) is somehow just a URL and a title, plus some random metadata (favicon and date, for example). The World Wide Web is not a book. It changes and its information tends to morph and disappear.