While I don't mean to second-guess your plan of working toward GCSEs, home education does open up a great many alternative paths, and it may happen that your son will thrive on one of those. Though it's true that the exam route appears to be the most popular one, certainly in our circle, my teen and a few of her friends have gone straight into work without any qualifications. Home education is incredibly efficient, so it frees up plenty of time to dabble in part-time work, do some volunteering, talk to adults doing various jobs, and learn practical job skills. This can make HE kids an attractive prospect to employers.
It's also worth remembering that there is no particular age by which exams must be finished. Even now she is 18, my eldest hasn't ruled out the possibility of doing some exams at some point; she just hasn't done them yet. The school system is very rigid in its expectation that GCSEs must be finished in Y11. What's so special about that age? We don't expect everyone to do their driving test at the same age, or predict that they'll never learn to drive if they haven't cracked it by 18...
Admittedly, kids who have spent many years at school may have absorbed the message that they only get one chance to do well at GCSEs, and doing otherwise means they'll fail in life. And if they believe that, choosing a different path is unthinkable and demoralising. But the more time kids spend associating with adults and young people who don't share that belief, and who may have been successful in life without the expected set of qualifications, the easier it is to keep their chin up if they can't succeed at exams or choose not to take them.
Another point is that if GCSEs do turn out to be the best route for him, you can be very flexible in how you approach them. Academically capable children often take one or two exams at a younger age than they would at school, then continue to do a few at a time. That is a very popular approach which allows kids to focus better than they might if doing 8 or 10 or 12 subjects simultaneously. They can gain confidence as they go on. If they are particularly talented in, or excited by, a specific subject then they can progress to an advaced level in that subject while still doing other subjects at a lower level. Some very bright kids don't bother with GCSEs, but go straight to A-levels instead. There are no required subjects, so he can do whichever ones he wants, or the ones you think he needs.
Anyway, your son is only 11 and by the sounds of it he is very stressed at the moment, so if I were you I'd relax about the academics and keep an open mind for the next year at least. There's no hurry. Let him have a break while recovering from school, and explore whatever interests him. This buys you time to look into different curricula and resources on offer and see what may be available to you locally. You could then introduce subjects gradually, starting with his favourites, so he gets off on the right foot and feels positive about his education.