You said, "Some DM's simply don't care. If the DM does care, they'll let the players know. I know I check over all the characters before we start playing when I run my home game. I'm pretty lenient on backgrounds, but if you use a custom background I at least want it to make narrative sense."
This is at odds with the rules for custom backgrounds, which simply say, "To customize a background, you can replace one feature with any other one, choose any two skills, and choose a total of two tool proficiencies or languages from the sample backgrounds."
There is no requirement within the rules that the feature, proficiencies, or languages make any sort of narrative sense. Most DMs, I gather from this thread, recognize that there should be an additional requirement that it has to make sense; so they house rule in DM oversight, to prevent abuse.