It's a complex issue. But my overwhelming feeling both as a mother and as staff is that it wouldn't work in the current set up of many hospitals. It would only work when the entire ward was single rooms.
I've had too many difficult situations such as men provided with a chair, who have then been found either in the bed with the woman (unacceptable for reasons which include infection control, the woman's comfort and recovery and preventing staff from carrying out their duties by being in a hospital bed!!) or in the bed on their own with women in the chair! Which was just unbelievable! 
Men who have been told to remain in the single room have been found wandering around the wards, helping themselves to patients tea/coffee facilities, using bathrooms which are for patient use, going in to wrong rooms, buzzing in and out the door all night going out smoking etc.
I have to say, women having a female companion would be of much more use. Although that still gives the same issues with extra bodies on already over cramped and understaffed wards. 4 beds, 4 cots, 4 extra stayers. That's 12 people in a bay which would house 4 on the general side...
I do think having support is important, but not to the detriment of everyone else. At the moment, desperate underfunding means that this wouldn't be possible in many places without infringing on the rights of other people.
OldLady most mums are fit for discharge fairly soon (within 12 hours) of birth. Most extended stays are due to the baby needing observation or treatment rather than for maternal reasons. So i think your point would be irrelevant in most cases. Even women who have had a c/s can go after 24 hours.
Although it's worrh mentioning that sometimes mums are readmitted with complications. A lot of mums chose to leave their babies at home with Dad unless they are breastfed. So babies can be left at home when mum is on the postnatal ward.