I think how it works is:
The Head has to agree
They get the funding for the whole child (as 'twere) and they remain responsible for the child's education.
I think this often means that you are expected to cover at home waht they do in school (but of course it takes a fraction of the time at home)
and it's usually mornings only or home on Tuesday and THursday or something, so that's why they like to keep it in step with what they are otherwise doing in the classroom, because otherwise the child misses things which are the logical link between what they did on Monday and what they'll do on Wednesday.
It's not quite the same as HE, I think it's even called something slightly different legally.
But it would still give you and the child much more freedom. and might well be a good stepping stone.
Some Heads are very resistant to the idea - you'd need to approach it in the right wide-eyed and grateful-for-all-the-wondrous-things-you-do manner probably, because otherwise, if you think about it, it automatically exposes the inherent time wasting of the school environment (with the best teachers in the world, there's still going to be all that in betweeny time of getting everyone sat down and listening with the right work book in front of them, yk?)
And the silly thing about all this is that in the olden days, when schools had more self-governance, this sort of arrangement was not unheard of at all, and the primary teachers would be able to respond to the individual needs of a child without reporting a family to a truancy officer. My mother clearly remembers the day when I was about 6.5 when my primary teacher said "gosh, do you realise she just did her first full week?" because before that I'd always be wiped out by the end of the week and need a quiet day at home on Thursday or something. I wasn't "falling behind" or struggling academically or socially, and the teacher could use her discretion, knowing that what I needed was a day off. Very much harder to do in that casual way with national curriculum, government targets for attendance etc etc etc.