OP, I'm a grandmother but I agree with you. I am very careful to avoid being a nuisance to my daughters in law - but then, I have a life of my own, so it isn't hard. I'm not sitting around waiting to involve myself in their lives. I don't feel entitled to push prams or hold babies - I did all that, and loved it, with my own - so I do what my children and their partners find helpful. Quite often, that does involve pram-pushing or baby-holding but it's their call.
I will never forget those early days with my first baby when my otherwise pleasant mil wanted to visit all the time...it was hell. I felt as though my home wasn't my own. I still hate it when people just "pop in" - I'm actually very sociable but I dread being caught on the hop.
At that early stage, I just wanted time with my baby, time together as a new little family, and time to see my friends. What I had instead was parents in law who "popped in" every single weekday evening, after work, while we were trying to have dinner, get the baby bathed and fed, and settled. Oh, and we were "invited" - but it was a summons really - to their house every Sunday afternoon. On the odd occasion when we tried to break the pattern, they would be huffy - it was a constant source of stress.
And then on her day off, mil expected to spend the afternoon with me, no matter what. It was no help - I didn't want to off-load my baby, I was just learning the ropes myself and I certainly didn't need someone watching me breastfeed while she repeated that bottle feeding would be easier because then "other people could help".
In the end, we moved away. It became unbearable, we had to explain our every move, and they refused to take our wishes into consideration - it was what they wanted to do which counted.
Don't listen to posters who tell you you're spoiling a potentially wonderful relationship with gps. Our relationships improved all round once we moved and as the children got older, they were very close to their grandparents.
You have a mil whose controlling tactics already involve taking to her bed "ill" when her son doesn't pay attention to her - I'd be very very careful to nip that sort of nonsense in the bud.