I'm single, so this is all "what if" (although I do have a son...)
I work a full-time rotating shift and often have a week or two off at a time (when I'm not working 469 hours of OT in a year). Currently, I live in town, but will eventually be moving "to the middle of nowhere".
Reasons it would be handy-- on the days I don't want to venture out (will be more common once I move), if I knew someone was going to be coming home early, and could stop by somewhere they might not usually be able to stop (during Post Office or small shop hours), I might ask if they could stop on their way home for me. Would save me a trip and the other person would still get home early.
I'm not one who is going to get "caught" doing anything wrong, but I wear a lot fewer clothes when my son is not home. If he came home early and walked in on me in just my underoos, well, that may be a bit embarrassing, so it's always nice to know if he's going to be home early. Now, if it was a spouse, that wouldn't matter at all.
My schedule is so weird that I have to keep track of it with a calendar. My son doesn't have the calendar, so if he cares if I'm working or not for any reason, he'll ask. If I get a chance to leave early (rarely happens because we have to cover the shift 24/7), I might give him a call and ask if he needs me to stop anywhere on the way home. If I don't feel like stopping anywhere, I just come home, no "warning".
I've had many coworkers decide to leave early and they have caught some people in the act (a spouse or a kid with a "friend").
Oh, for a babysitter... When my son was young enough to have one, if I got off a few hours early, I might have gone home to take a quick nap, or tried to do my shopping by myself, etc. However, if I was going to pick him up, I would give her a call. She did daycare in her home, and it was always easier when a kid was "prepped" to go home, so I didn't show up right while he was in the middle of something fun that I'd have to tear him away from. I never felt the need to catch her in the middle of anything, it was just easier on everyone when the kid got at least a few minutes heads-up.