As a MIL, and a GM, I gave my D & SIL space when they had there first son 16 years ago. I was actively involved with caring for him but at first, could not take him out on my own. A friends intervention changed what was an over protective reaction rather than any concerns over my ability to care for him on my own.
2nd son arrived 4 years later and they were both more relaxed. I still gave them the space and respect as parents to know what's best, according to current trends, often through gritted teeth.
Ds 1 & 2 followed and by now I'm the sole surviving GP, D & SIL are much more relaxed with their responsibilities and we have a good, close relationship.
For the OP, my daughter never let me take No 1 son out on my own for around 4 months. Until a friend asked her one very relevant question - As a mother did I ever intentionally hurt, starve or push her/him under a bus? Did I ever neglect her/him?"
Your MIL raised your son to be the man you fell in love with, married and had his son.
Given that it Is your First, I assume, and you are feeling a tad overprotective, what has she done, apart from offering "out of date advice" that causes you to be so hostile?
Why not invite her to a 'play day' at a local park and let her play with him, under your watchful eye. Meet for a coffee, lunch or tea, somewhere neutral, and let her push the pram. Then, when you realise that MIL isn't the ogre you've painted her, let her gradually care for him on her own while you relax.
Or if she really is an ogre, at least you've tried to include her.