My understanding of an EHCP is that it is specific to the educational environment, so the LA wouldn't consider themselves bound to deliver all that material assistance if your child were removed from school. I might be wrong, but that's my impression.
There are many many home educated children with SEN of one kind or another. Anecdotally, once the stress of the school environment is removed, and they've recovered from that experience, they begin to thrive and not need many (or even all) of the interventions that were required within school. Also, when home educating, the parent can really work at providing social experiences that are at the right level for that child, and that are managed so that they are positive, while the child learns the skills to begin to navigate social situations successfully and independently. And of course there are lots of home educated children who aren't maybe functioning socially with the confidence that one might expect of their age group, and all of their parents helping them learn to navigate with grace. I'm a real advocate for home education as a way for the bullied, the awkward, the out-of-kilter to learn to be socially confident.
I don't know of home educated children who are accessing specialist educational support, but it may just be that I don't move in those circles. I would say, though, that quite a lot of that support becomes unnecessary once you're working with the very high adult to child ratios you find in home ed, and once you can detach from the school mindset of educational product that looks like everyone else's educational product. E.g. if a child has trouble with literacy, you can do lots and lots of their education through conversation, and through you scribing for them or reading to them. Through that, they'll pick up literacy without stress as they go along, and you can get advice from other home edders about good approaches to literacy when there's dyslexia in the mix. And when they become independent with literacy is no big deal, because nobody needs to have paper evidence of their learning - you know perfectly well what they are learning, because you are there having the conversation, yk?