Mother of four boys here and if I am lucky enough to become a grandma and mil I will wait for my dil to invite me. I won't be insisting on visiting rights to grandbaby when she has just given birth! I will understand that she may need tine to recover, that she is likely to be feeling sore and fragile and hormonal and as such vulnerable.
I will be happy to offer food deliveries of home made stuff etc but basically I will respect dil wishes about what she wants in those emotional filled days after birth.
And no I wouldn't sit at a hospital whilst anyone was in labour my own dd or a dil unless the WOMAN IN LABOUR specifically asked me to!
Yes the father may need support but no he doesn't get to insist on having someone at the hospital to support him. The woman is the patient, she decides on that. If the man needs support he can phone someone etc.
Fwiw after ds1 my mil and other relatives of dps came to hospital first, his mum was lovely. Brought me a gift and that was fine, yhe other relatives whilst lovely were people I hadn't really met before. I was in considerable pain due to spd,and stitches etc trying to bfeed and basically found it overwhelming. I excused myself to go to the bathroom and a midwife saw me crying, she knew straight away what the problem was! And asked visitors except dp to leave so I could rest. Only they insisted on taking dp with them for the rest of the say, out for a meal and shopping etc, so I was left on my own.
With no 2. Mil watched ds1 and came to hospital bringing ds1. Once home same relatives turned up asap and stayed til late at night.
With ds3 my dad watched other kids, he saw ds3 first as I came home asap after birth. We then asked other visitors to wait a few days, it was Xmas time as well. My mum and mil waited few days and then cane and we're fine with that.
Ds4 a friend watched kids, I don't remember who visited and when!
Dd my mum watched kids and nipped to hospital very briefly to see Me and dd. Other family came once we were home over a week or so?
Labour and early post natal period are about the mother,she has just gone through birth and is recovering. Her needs are paramount.
If my dp was ill the idea that I would demand to have support at the hospital and once he was home recovering... Laughable. We would do what dp needed to feel most comfortable.