She didn't answer my question about *why* her husband didn't want the mom at the hospital. She said in her reply to me that she was glad to see her mom after having the baby, something her husband tried to prevent from happening.
You should try to side with your spouse but that goes both ways. If OP has *no* issue with mom and mom has never wronged OP's husband or treated him poorly and OP doesn't want her relationship with mom to deteriorate then she should have the conversation with her husband about what it is about mom that bothers him and that makes him not trust her with their son. If there's a legitimate issue then they will see if it can be resolved or at least they'll have had the discussion. If it's just his pride, he's the one that should change.
OP said that she thinks her mom coming around and offering to babysit is normal. She posted that she thought it was nice that her mom offered to help but her husband goes crazy over what OP considers to be a nice gesture. That's worth discussing instead of just letting him decide what is and isn't okay in their relationship and in raising their child.