I've already tried copying the "voice type" value from one sim to another in SimPE, it didn't change anything.

No, voice is determined by age and gender. There are multiple possible voices for some age/gender combinations, that's what "voice type" controls.

Is there a bhav or some other function that determines what age/gender gets what voice? Maybe that's something that could be modded?
Also, aren't there some NPCs with custom voices, like the Grim Reaper, Therapist, or Mrs. Crumplebottom? What is responsible for giving them those voices? I looked around their data and associated bhavs in objects.package and found nothing.

The voice type is part of the person's GUID, essentially a random number picked at his birth. You can flip the bottom bit to choose one of the two possible voices. It's a 16-bit number but only 1 or 2 voices exist within each age group. If you want other voices you have to assume another age with everything else that comes along with that, which is mostly fine considering that teens/adults/elders can all do most things, and the "clothes age" and "simantics age" remain separate. Young adult is another option. Special NPCs only have a few voice recordings for their appearance. It's not a complete voice type that can handle daily activities. I think the sounds are linked to the anims they execute.

But what about Ms. Crumplebottom? She has a non standard voice but you can do most of the normal social interactions with her, and she uses the standard animations for those.

Mrs. Crumplebottom's voice is probably hardcoded to only be for Mrs. Crumplebottom.