Yes, as a second time mother I coped - I would have coped better had DP been there (there was a lot of bullying). I coped because I'd already had a baby, and knew what I was doing, and had an inkling of what would happen if (when) it all hit the fan.
A solution then, that helps some women at least - my hospital had 2 post natal rooms - 1 was c-section/high-needs, 2 were standard. Let women choose. Sure sometimes choices will be forced (as they are no if you don't meet criteria - I wasn't allowed to even pay for a private room as I was post-c-section).
You don't need any more than a chair (which they have anyway, even in my crappy hospital, so I had somewhere to sit and feed). If most women won't have a partner with them for whatever reason then out of the 4 people in my room you'd only have 1 or 2 partners in the room anyhow.
Fine though, don't make it a right, let hospitals choose for themselves. Again in the children's ward there were official visiting hours, rules for numbers of visitors, but they weren't enforced unless someone was being a nuisance. Do that in the post-natal.
Let the market decide - let people choose if they want to be on a ward/in a hospital that lets partners stay or not.