I think he absolutely does get it...that's why the prospect of knowing that he'd at least occasionally have to step in to a short-notice illness day, INSET, or half term doesn't appeal & he wants to keep the luxury (all his!) of the status quo.
While I agree the logistics would need strong teamwork to make it work, the mental health & self-belief & sense of accomplishment & social interaction & societal value benefits of you doing at least SOME paid work would be of enormous value to you. It sucks that the legwork of positioning this with DH as being of benefit to your whole family unit, long-term falls to you OP! He will have plenty of responses that revolve around the fact it might make his (currently unbelievably cushy) existence slightly more hard work.
I'd start with the fact that having a wife who is going bonkers/getting depressed isn't a healthy family dynamic - and that feeling more like your old self is likely to improve your relationship as a couple. And maybe follow up with the fact that if at any point he developed a chronic health condition or cancer, wouldn't it be nice to have some earning capacity remaining in the household to help carry the load through his turn for getting the shitty end of the stick...
It sounds like generally things are ok with your DH, but the scenario he prefers vastly increases the probability of him wandering elsewhere. Far better to find something to ramp up your self-esteem & earning capacity once you've done the hardest years in childcare!
Can you have a short-term family crisis that necessitates you visiting a relative alone for 2-4 days?!? Ideally a weekend + at least one working day. Having to step into your daily reality might be the kick he needs to acknowledge quite how gilded his current existence is!
Given your career background, would a part-time postgrad course be an option for you for now? A background in childcare development, plus your SEN experiences give you a strong position to specialise in an area of the education sector which is wildly understaffed at the moment. Would remote study to work towards being some kind of SEN therapist/counsellor be of any interest? Maybe something where you could work part-time hourly and set your schedule in due course? Or aim for a school based role which could be term-only?
I hope your DH gets over his empathy bypass, wherever you end up! 