No, a 6 month old baby doesn’t NEED grandparents, but who actually wants to prevent a grandparent who loves the child from spending time with them, barring actual concerns?
(a) The OP is not preventing the grandparent from spending significant amounts of time with her baby! In answer to your question, I would agree 'yes who would stop this lovely thing from happening?' but the OP isn't stopping a relationship from developing at all (spending time alone together is entirely unnecessary for the development of 'a relationship' (ahem) at this stage). This would be loads of time for any grandparent and is only deemed insufficient because the MIL in question has set up a nursery and clearly saw herself as a primary carer and needs a job or a volunteering opportunity
(b) The OP has concerns. From MIL's disgusting behaviour (if someone had mooed at a breastfeeding mother in front of me I would have asked them to leave) and the fact she doesn't give a shit about her DIL feelings it is clear that she is not likely to respect her directions. I have also encountered the kind of MIL who asks her DIL something and then asks exactly the same question to her son when she didn't like the first answer. It's indicative of someone who will play the circumstances to get what they want, not someone who respects a mother's guidance. That, in tandem with the poor attitude already displayed to the mother of the baby, is enough to be a concern for many people.
(c) If the mother of the baby is concerned, there is already a problem whether she's reasonable or not. Because the baby (and this is something bertrand as a mother of boys does not perhaps pick up on) comes first and needs a relaxed, happy mother, not someone who is tense, worried and gritting her teeth after a 'break'. That's not good for a baby and the potential drop in mothering is in no way outweighed by 'bonding' with a middle aged relative at this stage.
(d) The way she has set up a nursery and is so keen to get what she wants to the point of putting unwelcome pressure on her DIL makes it clear she's looking forward to playing with the baby more than meeting the new family's needs. Understandable enthusiasm but there is no place for pandering to it if you don't trust the child person applying the pressure. Setting up a whole other nursery with no encouragement from the DIL is worrying, especially when you're dealing with someone who is already manipulative. It seems like the MIL has a plan for how she wants this to be and that comes before respecting very simple, reasonable desires in other people, such as wanting it to be a given that you will be taking your own baby home in your own car at the end of the evening, with no pressure to act otherwise. That is ridiculous behaviour and would make anyone back off.