I wouldn't hand-curate the tags, but parsing a well-commented HN discussion, or the fine article under discussion, might lead to hints. For instance, given a Title term "Matrix", one might parse for "movie", "mathematics", "transformation", "neo", "algebra", "agent smith" and discover to which tag "matrix-movie" or "matrix-math" the discussion should be assigned.

Adding a UI that presents the tags in the Algolia search might be the difficult-to-get-right part.

I guess classification via combination of article and comments shouldn't be too hard (at least in my imagination). I would like to subscribe to tag-based RSS feeds then, ideally.

Did you just invent subreddits?

No, why? I'm talking of automatic classification with multiple tags per post on top of a random mix of topics as we have it here.