Tried messing around with it again after updating to PC Patch 1.118.242.1030. The pets/horses still disappeared from the relevant households pretty much immediately upon starting a new game (they were in the thumbnails but gone when I tried editing), but it looks like if I replace them with saved versions from before and keep those versions in My Households, the animals stay? Although that still leaves out Mayor Whiskers and Captain Whitaker since they can't be placed into the world on their own.
The only way I've been able to get Mayor Whiskers and Whitaker back is to place them from the gallery into a household with a throwaway Sim. Move the household into a lot, go into live mode, then use testingcheats to shift-click on the throwaway Sim and choose "remove Sim from household." They'll disappear into nothingness. Immediately go back to manage worlds, evict the household from that lot, then manage households. You should see a "currently not in world" household with only Mayor Whiskers or Whitaker remaining. Unflag them as a played household so they can go about their happy business.
(I really don't see why we can't just create pet-only households as strays in the first place.)
The downside to this method is that Whitaker loses his ghost form. It's the only work-around I've found so far, though, so it will have to do for now, since these characters are important to the lore of their pack.
Honestly I kinda liked the unghostified bug with Whitaker since I'd always make him unghostified in gameplay anyway, but I can understand folks wanting the classic ghost version 🤭
