> I don't know of any service that satisfies all these requirements yet (although I'm open to suggestions, of course)

Former delicious user here too. I use Pinboard [1]. It's not free but has been working great for me for years. Has an API, extensions, bookmarklets, a 'toread' category, public/private bookmarks and really good tagging.

[1]: https://pinboard.in

[ Not affiliated in any way, just a happy user ]