Well, I started the new family, and tried the tips and tricks in this thread again, and imagine my surprise:

My family goes to bed at night.... often on their own!
I vary their activities (as I did with my old family), and they do not get tired as fast as my old family!
When they are in bed, and I turn the game off for the night, and I come back in the morning and check, they are almost fully rested!

Now I suspect something weird was just going wrong with my old family.