> The obvious solution is to add full-text indexing by page content. And maybe even store pages for offline use.

I built something that did exactly this, over ten years ago! I was a much less capable developer then, and it never worked as well as I wanted, so it was never truly finished.

Today this would be even harder, with the prevalence of web pages that are actually _heavily_ dynamic apps. Navigation to a new "page" is a much fuzzier concept now. What gets indexed? How do I generally know what content is/is not addressable by URL?

> And filtering open tabs by a keyword.

https://support.mozilla.org/en-US/kb/awesome-bar-search-fire...

"Add % to search for matches in your currently open tabs."