Did you even click the link provided? It provides some pretty good reasons with examples and a video demonstration of why it's so important. I don't know anything about your ex wife or what your living situation is, but maybe show it to her not as an "I'm right you're wrong" thing but as a genuine safety of your kid thing.

As for your child screaming bloody murder, there comes a time when you need to realize you're the parent and they're the child. For obvious reasons they are not equipped to make decisions regarding their own safety, which is why they're supposed to have you. Sit down with and have a conversation with them Mr. Roger's-style (meaning don't treat them like they're an idiot), maybe try and make a game out of it. If they still scream then honestly too bad, let them cry it out. You don't have debates with your child about if they can stick a fork in an outlet or not do you?