I've always been fascinated with web clipping – i.e., using some sort of extension to clip a screenshot, chunk of text, or image from a website. In fact, one of my first startup ideas in college revolved around social web clipping (hint: it didn't work out, despite having the awesome name snippus).
The most successful web clipper to date has been Pinterest, which created a curated image database of web clips. For text, Evernote arguably used web clipping as a go-to-market strategy for acquiring users. The logical successor, Notion, also ships a web clipper. These extensions also (unfortunately) usually collect your browsing data (if given the permission).
Rap Genius (eventually just Genius) tried to ship their annotation technology to the web. You would highlight a snippet, make a comment, and then it would show up for others. It was an interesting layer that sat over sites but never found much traction.
The lack of success in web clippers may be because of Chrome's locked-down extension policy. It is hard to balance a browser security model that lets users inject clipping code into arbitrary websites without having some sites use it nefariously (for tracking) – see the programmable web. Chrome eventually shipped a tangentially related feature, which would highlight a phrase that you searched for on the target site.
Maybe screenshots as an API can bridge the gap on some of these problems.