@JudgeJ "And hopefully this precious family member will understand if, in the future, she is seeking some help from the OP if she is told No!"
The baby is not a bargaining chip. How utterly selfish to imply you wouldn't help if asked, just because you didn't get to see the baby when you wanted to. This time is about helping the parents establish themselves in their new role, to figure out how it all works as a family of X.
OP, I understand your questions, and I think it's great that you've asked for help to understand this. It's actually brought up some memories for me that I'd prefer not to think about. I had my twins in May this year. In my MIL's need to "help", I was left feeling like she didn't even want me in the room with my babies. She was very insistent, even though I was and still am struggling with my mental health as a new mum. I felt like the incubator, and as I'd done my bit I was expected to step aside. I didn't want to leave my babies before I was ready to, I needed help to facilitate me staying with them. That could have been making cups of tea for me, making sure I ate something, help with putting washing on or cooking us a meal. She did none of that, I ended up making her tea! Everyone wants to hold the baby, but I can do that, it's all the other things I needed help with. Only my sisters understood that, which I am so grateful for. Establishing the bond as parents is so important, and should be paramount. And it was infuriating that my MIL would insist she was there to help, but wouldn't do anything apart from hold a baby.
Just give them space. Let them know you are there to help if and when needed. Ask if you can make them a meal and drop it off. Be there for them, but give them space.