Does the AI have access to a general-training building?

[TinCow]

But wait. If the new generals are just generals, and not Family Members, won't they just become rebels when you kill the faction leader? Or are these new generals listed as Family?

If they're adoptions, they're family members. Adoptions are the most common thing when you're low on family members, so most likely that's what the AI is getting as well.

If you want to see this become REALLY annoying, you need to play RTW:BI. The horde factions in that game often get whittled down to a stack of only family members. If you let just 1 of them escape, several more spawn in and keep them alive.