On the subject of what they do besides engagements, I agree - it's unlikely they're running around trying to get everything done, but I think there's more to an engagement than the hour or two they're there. Lots of planning, briefing etc. And I think that while their staffs probably handle a lot of the details I always get the sense that the royals approve every plan. Plus for the women, the sheer effort of being appropriately dressed, made up, and generally beatified has to take a lot of time. Personally, I'd hate that element. I hate having my hair done and the thought of being absolutely tied into a trip every 6 weeks, colour the same and a professional blow dry at least a few times a week leaves me feeling awful.
I think there are also a lot of things we don't see. Sometimes we hear about something afterwards because someone leaks it or posts a photo or something, but I think there's a lot more of that. I do remember reading an article with an interview by someone who was running a charity with a royal patron (can't remember who) and she mentioned that said royal was far more involved than just in the public stuff - convening meetings with others who are interested in similar issues, chairing board meetings etc.
I can easily imagine that they "work" at least a few hours a day, plus engagements and then get time to do their own thing/ spend time with kids between. Certainly, I'm guessing Kate, unless she has an engagement, is on the school run and with kids in the afternoon/early evening so ensures any official work is done in the mornings while kids are at school.