"The MIL has successfully raised at least one child to adulthood, while the GF has been a mother how long?"
I don't think the last part of this is necessary. The GF has been a mother for 18 months and has managed not to scald their son during that time. The benchmark of good motherhood doesn't increase by each day you do it and manage not to kill your child.
The GF has had a terrible shock and could be dealing with the repercussions of their sons injury for months and years to come if the injury is as terrible as the one Keema described. And I mean dealing with it in the most graphic and hands on way, if she's going to be the main one changing dressings while their child screams in pain. Doing that even once sounds horrific for all concerned and Keema my heart went out to you and your child when you described having to do that.
Ten weeks to come to terms with a shocking, severe injury is nothing, especially if she feels unsupported and attacked for wanting to stay with their son all the time. It doesn't have to be about punishing MIL or not trusting the OP with their child.
It could be about the unrelenting and irrational but no less real fear of what might happen to him while he is out of her sight and guilt that she wasn't with him when he needed her.
And I'm sure she realises this is irrational and that something could still happen while she is by his side. But that still doesn't mean she will find it easy to relax, to let the fear go or to feel comfortable about him returning to the place he was injured without her.
OP if your girlfriend is as strong willed as you say, pushing her and fighting her are not going to help. It may be that you and your mother have to give her more time to calm down and come to terms with what has happened. Because what happens when you push someone is that they take a step back, and then another and another, and nobody gains any ground. I'm not saying I agree with her, but you and she, as the child's parents, need to talk about this properly between the two of you and not in terms of what your mother wants or feels.
MIL managing to raise someone to adulthood is neither here or there by the way. The criteria for being a good parent doesn't begin and end with how old your child is and if OP's mum is a better mother than OP's girlfriend it's not because she started 30 years earlier than the girlfriend did. By that reckoning, I must be a terrible mother to DS since my two other children are dead.