It probably wasn't a private residence either, consider this:
>The local synagogue probably kept a flock of sheep for sacrifices
>That synagogue would have building support for keeping that flock, including an animal birthing room
>Jesus being born in such a place would be another connection to his typological relationship as the sacrificial lamb
Here's my proof question:
How could the shepherds to whom the angel appeared make with haste and find the newborn baby if he was at one random house?