Wow! It's not even just sexism - he's NOT EVEN doing the traditional "male" tasks (DIY, financial management) does he even wipe his own arse?? 
I thought my ex was bad this is on a WHOLE other level. Sorry I would STRONGLY advise considering leaving because even if this person improves I bet they'll STILL be doing a lot less than you.
My ex did - all the DIY, car, garden and house maintenance, cooked at least once a week and cleaned up after (basic like stir fry/pasta but not as basic as beans on toast), financial management (kinda his job so he was good at this), his own ironing (army I couldn't do it right however hard I tried, so one day threw it at him - ok YOU do it), put the bins out.
He was however apparently incapable of laundry, hoovering, dusting, cleaning windows, cleaning kitchen or bathroom, cleaning floors, grocery shopping, childcare (even though he now has 6 kids), remembering to empty all the bins round the house before putting the bins out.
Wish I'd found mn before I married him partly (but then I'd not have dd).
Yours is far lazier than ANY man I know and that's saying something!
Annamagnani - that's not necessarily a man thing I have a dear friend who REALLY can't cook - I'm talking struggles even with oven heat up meals! She's blown up 2 cookers and a microwave and ruined countless pans even kettles (forgets to fill them) trying to learn. She's intelligent and practical (lawyer) Mother and sister are chefs but she seems to just be honestly incapable. 
Angstina - your friend sounds like my ex a bit - what he DID do he wanted a fucking medal and ticker tape parade for (even if he did half a job).
I also know women, even wives/live in gf who can be lazy but not this bad. Even a very lazy flat mate I once had.
I think it's likely you're not so much depressed as bloody knackered!!
"His 'issue' was that he genuinely believed he was rubbish at housework and that he'd rather not do it at all than risk disappointing me by doing it wrong or not good enough." That's strategic incompetence Lazy women do it too sometimes "oh I couldn't possibly change a tire/hang a picture/phone the bank to sort my overdraft myself! I might break a nail!"