I used to teach in a SEN school. Most of the children had autism but some had other needs ranging from ADHD to MLD, and there was one class for children with more severe disabilities as well. The majority of children didn't sit GCSEs, but those who were considered academically capable were entered for the foundation tier and usually achieved a grade between 1 and 4.
The good thing about SEN schools is that the ratio of staff to pupils is so high that it's much easier to differentiate by ability than it is in mainstream, so he could be given some one-to-one support to go beyond the work the other pupils are doing. On the other hand, the pace of lessons is obviously much slower. We had one child at our school who was very bright but had severe ADHD (no learning difficulties at all), and the school were looking into whether they could be transferred to mainstream because it just wasn't feasible for us to provide them with the level of education that they deserved. That's not to say that all special schools will be like that, but it was the case at ours.
That said, and I don't mean this in a rude way at all, how confident are you that he would be able to achieve good GCSE qualifications through homeschooling? Most secondary age children, including those in special schools, obviously have several teachers who are each specialists in their particular subject area. Even as a qualified teacher I wouldn't feel confident teaching children outside my own field (science/maths). It's a lot to take on if you're planning on teaching the whole curriculum by yourself - I assume this is something you've already considered, but just wondering how you're planning to go about it?
With regards to children being allowed to wander the corridors, that would be very much dependent on the child and the circumstances. If a child was determined to leave the room we obviously wouldn't be able to physically force them to stay, but they wouldn't just be abandoned to roam around at will. A TA would go after them to try and bring them back, or to find some other way that they could do the work, for example in a quiet room on their own. If they were repeatedly leaving lessons it would be picked up on very quickly, and we would have a meeting with the child and parents about the reasons behind it and whether there were any strategies that could be put in place to help them.
A SEN school would be a great environment for him to develop his social skills while being provided with a lot of specialist support, and perhaps also a good way to help him see the positives in being outside the home environment after his difficult experience in primary. There should be a huge variety of non-academic opportunities provided as well, and in addition it would give you some breathing space.
reefedsail has good advice above re. considering what you want the ultimate outcome of his education to be. Do you think it is a realistic prospect that he might eventually want to go to university and have a professional career? If not, might vocational qualifications be a better long-term option than GCSEs and A-levels? A SEN school would certainly be able to help facilitate entry to a local college where he could be given continued support.