PS thinking about it more I think he’s manipulating you OP & using the I never wanted them card to guilt you into silence.
He actively participated in conceiving them. Like a lot of men they will get away with doing the bare minimum & he’s got the best trump card of all to use every time.
Then declarations of love etc to cement your feelings of being actually in a loving supportive relationship.
I know it’s not helpful to throw around armchair diagnoses but is he very rigid in beliefs, habits etc? You say obsessions. Asexual or not especially sexually inclined?
Was sex more frequent before kids? He may also associate sex with reproduction if it has resulted in something he says he doesn’t want.
V difficult but think you need to stop taking the blame for the kids & set yourself some basic non negotiable standards that you need - to continue.
Because he’s getting all the perks of your family life & you - love, support, company, stability but not contributing to it & your resentment will only grow.
Will he go & see the kids at nativities etc or are they too young ATM?. If he’s happy to do Disney Dad & all the nice gratifying stuff then he’s being a hippocrite.
You have every right to demand he step up - as a core requirement of your partnership going forward.
Would gr cope without you? Does he value what he has?