I agree with the others that she is lonely and she needs some attention. I don't think there is much you can say to her to get her to back off and I absolutely think that this is your husband's problem to deal with! He needs to develop a spine. At the store, he could have left you to deal with your child and corralled his mother at a distance to tell her to stop screaming. It has to come from him.
That said, I think you can deflect some of it gently - if the pediatrician doesn't want a crowd in the office, say that. Your husband can tell her that, after her squeamishness about ear piercing, he doesn't want to subject her to medical check-ups that involve shots and boosters! And you can say something more gentle, like, "I'd much rather you had fun time with your granddaughter than to have her associate you with doctor's appointments." Then make good on it - give your daughter time with Grandma, either without you or with you. MIL will get pretty tired of taking care of your child by herself, I think - but if she enjoys it, so much the better! Their time should be special and not associated with errands or appointments.
Be sure to invite her for birthdays, and see if you can find it in your heart to include her on Mother's Day. If not, invite her for Father's Day with her son. Make holidays about family togetherness and gatherings. Decide now how much you can handle - say 2 days a month? Then schedule them and enjoy the time in between.
I love the idea of handing the phone to the child when Grandma calls - as long as your child doesn't resent it. "Look honey, Grandma called just to ask about you! Tell her about what you did today!"
If she wants to attend a play group now and then, that's a little more social and maybe it's not so hard to include her if you keep her out of other things. If you think she'll use the time to tell all the mothers how to raise their kids, that's another problem. Unless of course they will support you and shut her down! One way to handle her opinions is to say noncommittally, "I'll consider your advice." And just repeat it every time, without expanding on it. Then change the subject, every time. Otherwise, if she says something off the wall, you can say to the group, "Isn't it interesting how child rearing has changed over the generations?" Just make it a light conversation, not a confrontation.
Don't expect her to check in on you - it's not going to happen, and it just gets more depressing to think about it. My MIL once called me in the emergency room, tying up the phone at the nurses' station, to tell me how upset my husband was that I was in the ER! He was in the city trying to get a train home, and she badgered the ER nurse into calling me to the phone to tell me about her son. Then she called the taxi company and made them radio the taxi driver taking me home to find out where I was. It took a few years but now I laugh about it and enjoy telling the story.
I don't know if volunteer activities involving children are a good idea for her - she may be the type of person who would be a nightmare for teachers if she went in to the schools as a volunteer reader, and if she's squeamish or a drama queen, she doesn't need to be volunteering in a hospital. I imagine that senior activities are limiting for her because no one there wants to hear her complaints. But if she is able-bodied, she might enjoy the distraction of some theater trips or bridge games with others. But that's up to your husband to suggest.
Meantime, try to find something she is good at and encourage her to develop that attribute for the benefit of others. As your daughter gets older, it might be good to include your MIL in any volunteer or community service projects you do as a family. Putting her in a crowd might work - if everyone is wrapping gifts for disadvantaged children at the holidays, or visiting a nursing home when your daughter's preschool goes to sing songs for them, or collecting for the food pantry, or picking up trash at a local nature area, MIL can be part of something that benefits others and that doesn't have anything to do directly with your parenting style, other than teaching your child values.
Good luck, and definitely throw this in your husband's lap.