Aside from some style tweaks likely to come around, the underlying technical bits I've been ignoring or had on the list to address (looking at you, tag views) are now in place and working as I'd expect. There's a pesky bit I've ended up handling in a more manual sense, at least for the short term: content view by tags.
Essentially I'd like the "categories" block results (those items on the right-hand side) to point at a contextual view for each tag, not the tag (taxonomy term) detail itself, because I don't care about the name and description of each tag.
I've found this article that feels like the closest match to what I'm trying to accomplish, but I've yet to be able to get this particular view page working as expected. I have both parts individually working as separate views, but I can't get them to play together (without borking up the whole tag/category list block).
So I've instead added rewrites for each tag, where folks are redirected from the taxonomy term link (the tag description I don't care about) to the appropriate tag content list view. It works, but it's completely manual process (to generate the rewrites) for each new tag. There's likely some magic with contextual filtering and relationships I need to iron out, but for now it works and I'm calling it good. It just doesn't scale well.
As simple as possible, as complex as necessary.
Headline photo courtesy of Meme Guy