AITA for not letting my grandmother hold our son

I (25f) am not allowing my grandmother (76f) to hold our one year old son anymore. When I told this rule to my parents, my mom (57f) said I was being rude.

Two weeks ago my mother reminded my grandmother of her doctor appointment, giving the time she would pick up my grandmother. In response, my grandmother said she would scratch my mother of she goes to pick her up.

Due to this, I told my mother of our (Husband and I’s) new rule for our son; that my grandmother is no longer allowed to hold him.
In response my mother defended my grandmother, saying that "It’s just the way your grandmother is. She didn’t mean anything by it. This is why I don’t tell you things, you always take it too seriously."

I personally do not think I am taking things to far. To me, I am not allowing a person who threatened their own child to hold my one year old son.

Should I just let my grandmother hold her and get over it?

Edit: Thank you everyone for your response. I will be reconsidering the rule after another discussion with my mother and make a decision with her advice. I also apologize for the wording of this all as I am not well spoken.

And to address some things:
1. I did not intend this to be punishment to my grandmother. I will explain this to my mother as that may be how she saw it as well.
2. I am more worried my grandmother will scratch my son if he does something she doesn’t like as this is not her normal behavior.
3. My grandmother may be strong enough to hold my son if he decides to sit still.
4. My grandmother has never made threats like this to family and does not joke about these things. She often scolds others for those jokes.
5. My grandmother does not have dementia or anything of the sorts.
6. My grandmother has had a history of violence, but it stopped about 5 years ago.