Well, I'm a terrible multitasker, and HE works for me.
I don't know whether that's because we use the autonomous approach and therefore the kids have responsibility for their own learning. I help them when they ask me to, but they always take the initiative.
There also seems to be a lot more time when you home educate, so you don't necessarily have to be efficient in order to get things done. I know a number of HE families who rarely stir out of the house before 11am. They spend the morning getting breakfast, putting on a load of wash, packing a lunch, and finding somebody's lost shoes. (I'm like that too on the days when I don't go anywhere early. It's pleasant.)
Consider that most schooling families have to accomplish all the same things before 8:45am, and that they can't just go late or decide to stay home after all, and that there's trouble if a kid turns up without all the proper kit. It's clear that parents of schoolchildren are the ones who need to be well-organised!
Anyway, that's what I found during the term when my older dd tried school last year. We managed brilliantly well (for us), but it took all my mental energy. Whether I could have got more than one child ready for school is open to debate!
I haven't read Free Range Education so I don't know if it paints an unrealistic picture of how much a disorganised, mono-tasking parent could accomplish. But I'm sure you can accomplish what you need to! And besides, while you are faffing around trying to figure out what's for dinner, the children are educating themselves. Or you can just get them to make the dinner or buy the road tax online: they have plenty of time for that sort of thing.