They did put 'important lore' in Hildibrand. Constantly. Like, every single Hildibrand storyline has entirely serious and interesting lore outside of 'haha funny slapstick', and it's not even subtle about it; I'd genuinely put it up as one of the real reasons to play Hildibrand. It can't exactly be world-changing, because ultimately it's side-story and the game's written so that it makes sense without doing them, but it really does a lot to flesh out the world we're in. In order...
A Realm Reborn: The big one is the tale of Sil'dih; we have the truth of that in a more serious and modern form now int he Sil'dih Subterrane, but it took most of the game's life before that happened, from ARR patches to EW patches. It's also got the first instance of an unintentionally summoned primal, which would later happen again in Stormblood with Susano. And if you want to count it, Ultros does give an early idea of what voidsent are beyond snarling monsters, and that same Coliseum storyline gave us more insight on what that place is like than we had at the time; the more Coliseum-focused stories came after that, in Stormblood. Oh, and also Gilgamesh, which brings in the Final Fantasy multiverse in a way that helps all the future crossovers make sense; that's all the same stuff that games like Dissidia do, but it's still good to tell us here since not all of us played it.
Heavensward: This one's a little tenuous, most likely because they didn't exactly plan to do it, but there's stuff about aetherology that you can get from this one. To a lesser degree there's Ishgardian stuff, but we weren't exactly starved of that in Heavensward.
Stormblood: Welcome to Politics Zone! It's time for a dive into how Hingan politics work that none of the other storylines focused on Hingashi have ever done, even now! Co-starring surprise Thavnairian stuff: this was to my knowledge the first storyline with a Thavnairian NPC at all, and it's not a small role, it really gives us some insight into what sort of power Thavnair holds, both as a nation and in its independent agents and traders.
Endwalker: This one I'm gonna put in spoiler tags because it's recent, and people might not have done it.
Welcome to the most unexpected crash course on interstellar FFXIV lore since Ultima Thule's fish. We hear about two different alien races here, the PuPu and the Mandervillians; and maybe you'll dismiss the Mandervillians because they connect directly to Hildibrand himself, but I'd encourage you not to, because that's a pretty good window into how the End of Days impacted the rest of the universe that we didn't actually have otherwise; we heard from the Ea that at least one other civilization succumbed to the same thing, but nothing further... and now we have that something further. (And personally I suspect the Mandervillians may come from Deneb III, but that's a different discussion.)
It also gives us a pretty solid window into how more 'common' people see Sharlayan, albeit a bit of a twisted one; we can't take the Thavnairian Truth as actual... well, truth, but he definitely shows Sharlayan's secrecy has led to distrust. Also, more about Dr. Lugae, which helps flesh out some of the political situation in Garlemald.