The state of maternity care in the UK is scandalous. As a service provided exclusively to women this does raise questions as to the reason. I suspect there would be an outcry if similar standards applied to an exclusively male service.
It isn't good enough and has never been good enough. Even in the interwar years the UK had a disgracefully high maternal mortality rate. You can't unpick a mess like that overnight, or with one petition, but this needs to be given far more priority than it's received so far.
In the meantime, and in answer to the OP, men as birthing partners are absolutely fine in the delivery room and the very well monitored recovery room immediately afterwards. This encroaches on the needs and care of no other patients.
On post-natal wards, I am against and would sign a petition to that effect. I had an EMCS and a frightening labour owing to DS decelerating: it was a very scary time until he was safely with us. Born at 4pm, DH with us in recovery room then out for the night by 9pm. This was fine. We'd have willingly booked and paid for a private room but none were available at that time. There were also no private facilities for delivery within many miles of our location.
An aside, as to how female-only care is conducted, those data-mining companies should be nowhere near vulnerable patients who have just been through a major physical trauma. Do they still let these people in? I find it a travesty that this ever happened in the first place. They'd not allow ambulance-chasing personal injury firms into a major trauma ward where people are being stitched together following accidents: why, then, post-natal women? I think the answer is obvious.
I'd want rid of them first, followed swiftly by men taking up female patients' space, washing facilities, etc.