Most of the disorders they test for are treatable and yes, you can have a child who has one of them and there was no indication from the parents or siblings' medical histories. I don't understand why you wouldn't do the newborn screening.
I'd vote for doing it. Here in Oregon, there's been a push lately to provide it to babies who were born out-of-hospital, so the state is starting to loan screening machines to midwives and birth centers.
We have moms make frozen pads (or we do it for them while they're in labor) and they generally love it.
I know the adult diapers sound crazy, but moms who try them tend to love them in the first day postpartum.
I thought I wanted to have my older children there in the house when I had my fourth baby. Midway into labor, I realized I couldn't really relax and focus on what I was doing with them there, so I had a friend come borrow them for the day and bring them back after the birth.