The board is slow, so I'm going to throw something out.
Well, DH has never SAH full time, but he is at home more than I am. Right now, he teaches an evening class twice a week, so I'm at home alone with the kids then. He also likes sleeping in on weekend mornings, so I'm alone with the kids then. Even if he didn't teach (or maybe especially then), DH would most likely have other things he'd want to do on his own in the evenings.
Growing up, my mom SAH until I was about 8 and my sister was 5. She didn't like to go grocery shopping with us, so my dad was alone with us then. She also had other things she did occasionally and we were home with dad alone.
I have to say though, that I'm home alone with the kids much more often than my dad was with us.
In my experiences, it's the sports/activities time where the wohp in the sahp/wohp partnership spends the most alone time.
When I was a SAHM on average DH was home alone on average about 6 hours a week while I exercised or ran errands.
He also took care of a lot of the kids doctors appointments.
Sometime when I was volunteering he was off work and was home with the kids.
My dh handles many of the appt.
Did you SAH once your kids were in school?
We are sahp/wohp but I am not the only one who arranges things. DH has done mainly the sports and scouting things. I do virtually nothing with scouts and DH is very active.
He also is in the habit of taking one or both of the boys to the library on the weekend. He is there if I have a nighttime meeting. He will frequently go for an evening walk with one of the boys. Things like that.
Even if I was a ftwoh, DH would not be taking the kids sick days as he already takes a FMLA day once a month to visit his mother. So he doesn't have the time to add to the that time away from work.
Sorry, but I can't remember your situation.
I think it's nice for a parent and child to have something that is their connection, something the other parent is not involved in.