I'm pretty sure that villagers go about their business independently of where the player is, but they do need to be in loaded chunks which means basically within render distance, in the spawn chunks or, I think, in chunks that have been set to be constantly loaded with the Forceload command.
I don't think it's based on render distance but rather a set value.
I know certain entities (iron golems and villagers in this case) only visually load for me and fully carry out their routines within a certain radius around the player, because I play with a render distance of 32 but entities certainly don't load out that far, and visually "pop in" only when much closer. I can step forwards and backwards in my city and they will appear and disappear.
I doubt it is only visually that they are hidden, as it would make very little sense to me that they are still carrying out their routine (as in, loaded and processing A.I. while simply being visually hidden and not rendered), so I'm fairly certain it's a predetermined value upon which they won't be loaded regardless of render distance.
I think there's even a "bug" where if your render distance is too low that mob spawning is impacted because they won't spawn within a certain radius of the player as well. On the flip side, they don't seem to spawn/render all the way out to the render distance though; there seems to be a cap.
I don't know what that value is. Someone who knows more would have to say (would guesstimate somewhere around 10 to 12 chunks, maybe up to 16?).
I'm pretty sure that the render distance bug is due to mobs outside render distance not despawning once the player gets past 128 blocks away and therefore keeping the mob cap filled.
Iron golem farms, at least, are supposed to keep running in the spawn chunks, no matter how far away the player gets.
Normal mobs spawn between 24 and 128 blocks from the player, become passive after a short while (and hostile mobs randomly despawn) when further than 32 blocks and hostile mobs immediately despawn when more than 128 blocks away (unless hindered by a too low render distance).
Villagers are special though and if they can "make" iron golems whenever they are in loaded chunks I don't see why they wouldn't be able to make babies as well, though I'm not absolutely certain.