I had a horrendous birth experience with ds1 in hospital: pre-eclampsia, induced, 3 changes of shift during my labour, 10 people in the room when I actually gave birth (don't even know who some of them were - the cleaners?), ds1 taken away when I got back to the ward and given a bottle without my permission, I staggered down to the nursery to see him when I woke up about 4am and cried because the door was locked and I didn't know which baby was mine, stayed in 8 days in a filthy hospital (blood stained bathrooms...horrendous), got no sleep, was told my a consultant I couldn't go home as I'd probably had a heart attack during labour and there was a risk I would have another one (left me sobbing over my soon-to-be-motherless babe in his plastic fish tank while he went off to terrify other women, presumably), was snapped at my a mw that if I wasn't going to do that "properly" (that being breast-feeding) I may as well give my baby a bottle, no-one came near ds1 for over 24 hours after he was born - they "forgot", and he got low blood sugar and no-one noticed, my canula was left in for 24 hours for the same reason, I was in agony with piles (couldn't walk, couldn't sit, could barely lie down) and it took 3 days to get any steroid cream for them...eventually got home and (surprise surprise) had severe PND.
I still don't think dhs/dps staying overnight in hospital is a good idea - I would have found it quite disturbing having men (I'll avoid the word strange) wandering around. Especially as I was half-naked, blood-stained, weeping, leaking milk and generally feeling extremely vulnerable. I felt similarly about all visitors, tbh - dressed, clean people from "the outside" were quite disturbing (I'll allow that PND played some part in this). But at least they went away sometimes - and I didn't feel so 'exposed' - there would be no respite from that if Dads were allowed to stay all night. And my experience is that women snore - double that on a maternity ward with partners - no-one would get any sleep.
My dh is a lovely, gentle, (mostly) sober, non-drug taking reasonable sort of a bloke. Not all men are like that. Some of the other mothers on the ward where I had ds1 were terrifying - let alone their partners. The security implications of having extra people stay over night are significant. And where does it stop? If it is your entitlement to have your dh/dp stay overnight, what about other children - what if you have no-one to care for them, do they come too? You can't have a rule that only 'nice men' can stay - the quiet, considerate and helpful ones.
Greensleeves' case is horrific. There should be some facility (sleep-over family rooms?) for people in those kind of exceptional circumstances.
Fwiw, I was terrified of going into hospital again to have ds2 - but my experience (different hospital - dh's hospital actually but sadly microbiologists don't have the same power that consultants do - we had to pay for an amenity room, he had to stick to visiting times) was much better.
And - even in the first hospital - most of the midwives were lovely - I'll never forget the one who sat in a little room in the middle of the night, listened to me sob for 2 hours and persuaded me to keep on with the breast-feeding - I think it was down to her that I managed to get feeding properly establised and I fed for 12 months.
Like every other profession within the NHS (well, every profession really) there are good and bad.
Maternity services are over-stretched (hell, they're closing the unit where I had ds2 down ) and changes do need to be made - but, as other people have pointed out, NHS funding is a finite resource and they are areas of greater need than providing space/facilities for fathers to stay.