You've posted a situation that is completely alien to most of us, followed by a comment that perfectly fits that situation. You cannot be surprised that most of us focus on the oddness of the situation as a whole.
Then you mentioned a medical condition most of us know nothing about and asked, 'in the context of this very particular medical condition, as it affects my husband, please differentiate between his behaviour and his comment (which perfectly describes his behaviour)'.
Having said that, there is a difference between 'I can't because of my medical condition' and 'I can't because I fundamentally believe that childcare is women's work'.
The problem is that the two things, attitude and condition, will have worked together, since your twins were born.
His belief that 'only mum can and should do childcare' will have increased his anxiety about looking after his children. At the same time, he will have used his anxiety as an excuse to avoid childcare, therefore not learned to do it and overcome his anxiety step by step, like any sensible person and devoted dad would have done, because he believes that childcare is women's work.
Medical condition aside, loads of dads adopt the 'helper' position to the 'expert mum' and use strategic incompetence to get out of childcare. There's nothing unique or unusual about that. Quite a few mums adopt the expert position for themselves and cast the dad as incompetent helper, thus trapping themselves as default child carer and martyr for years to come. Nothing unusual about that either.
If your DH was a great dad, he would have wanted to learn to look after his children and he would have made it happen, step by step, within his limitations. Becoming gradually more competent would have helped him gradually overcome his anxiety, so he could build up his childcare abilities.
He didn't do that because he believes that childcare is women's work.