If my suspicions are correct, then MIL’s probably grieving the fact that her relationship with her mom was a bust (for whatever reason).
I know logically, someday, when my son gets married and my DIL gets pregnant and tells me that she doesn’t want me in the delivery room, I’m going to have to suck it up and respect their wishes.
However, I would hope that my DIL understands that I just want to be there for her in a way that my mother wasn’t able to be there for me.
She has a primary need to want
to be there and
let you know that she’ll be there for you. “Being there” is a HUGE deal to daughters who lost that primary ‘mother-daughter’ relationship.
Find other ways to let MIL “mother” you, if you don’t want her in the delivery room. She knows that she will never replace your mom, but she wants to be needed as a woman and a mother. She wants to know that she still has an important role in the family and a place within the family’s circle of women.
Her sons don’t “need” her “womanly” knowledge in the same way that a daughter would need that knowledge. It’s complicated.
(I can totally see myself saying “but I had a boy” in 20 or 30 years from now. He doesn’t need me to swap labor stories and talk about “female stuff”! And he has his dad to talk about “men stuff”. He really won’t need me, like he’ll need his Dad.)
(If I’m wrong about where she is emotionally, all bets are off.

)
There’s a huge opportunity for your relationship with your MIL to grow and deepen as you go through this experience.
You are so blessed to have two amazing older women in your life who want to be there for you and guide you through this phase of life. Honestly, I would have been thrilled to have this “problem”.
My mom’s dead (died when I was 7) and my MIL is MIA (by my husband’s choice not to be involved in their lives and after meeting FIL and MIL, I totally understand why!). It is what it is.