No, they go food shopping, cook, put the laundry on, tidy up, pay bills, load the dishwasher etc.
Why would a man or woman come home from work and expect a partner who has been caring for children all day (perhaps with broken sleep) continue to do the stuff that he/she would have to do anyway if single?
Because a certain quantity of those listed chores can easily be done around childcare within the house. On the other hand, all of the listed jobs are impossible to do while working in an office, with the possible exception of shopping which might possibly be done pre/mid/post-commute.
To look at it another way, why would anyone WANT a partner and/or kids if it did nothing to make their life any easier? Isn't it a bit disingenuous to say of the person working "You should continue to do everything you would be doing if you were single" while the person who is staying at home looking after the children is flagrantly NOT doing so themselves? Because obviously if they were single, there would be no kids in the first place, and therefore they would be out working in a job somewhere themselves. Likewise, if person A and person B share a house, can't both of them say "You should be the one to do X, because that's what you'd do if I wasn't around anyway"?
Yes, childcare and broken sleep sucks, no two ways about it. But to suggest that it's completely impossible for a SAHP to (for example) pay bills at any point during the time period when their DP is working 9-5 or longer, plus commute, and that working DP should be doing that instead of them "because that's what they'd be doing if they were single" is pretty ludicrous. Likewise, if the SAHP is going to take the baby out in the pram at lunchtime (or the toddler to the park, or whatever) then isn't it entirely reasonable to ask them to put a little bit of shopping in the pram in passing, rather than expecting the DP to try to navigate driving to a supermarket and parking in the middle of rush hour at 5.30pm?
This "what if you were single" doesn't fly. Not for a second. Sharing a house is supposed to make your life easier and better. Not harder.