I think you hit the nail on the head when you said you discovered a core issue in MC but had no obvious solution on how to deal with it. IMO that falls squarely on the shoulders of the MC.
Early on in our marriage my H and I went to MC. Things were actually very good with us on many levels but we did have communication issues. We both thought it would be best to try and learn how to communicate better while things were basically good. IMO the MC we saw put us on a very dangerous path.
We saw her once a week (together) and once every other week (alone) for almost 6 months. All we really did in her office was discuss the same things we discussed at home. Her response was always the same... the two of you need to communicate better or the power struggle to be heard will never end. Um, we knew that and that is why we came to see you! She never gave us solutions, taught us any tools or anything of benefit. H and I were very young (25 years old) and I guess we felt MC was a good thing so we kept going even though it didn't seem like much was changing.
Like you and your H, my H and I had a core issue that we saw no obvious solution to. But, we all know there usually is as solution. I do feel for you as I know what it's like to go see a MC, find out a core issue and not have any guidance from the MC on how to solve the core issue.
Things then start to pile up. In your case... children, lack of attraction and continued financial strain. My H and I didn't have the child issue or lack of attraction to deal with but the communication issue festered and when big life events did happen (me getting sick, my H's mom getting sick and my dad passing away) things fell apart.
This is simply my speculation but it seems your H really doesn't have a full picture of what you do as a mom to special needs child on a daily basis. The regular care and feeding along with the homeschooling and all the diagnostic research. I know he appreciates you do take care of those things but I do wonder if he really *gets* how much work it really is as a single parent.