Brah one of my kids fell down the stairs a few weeks ago while my wife was trying to get out the door with them, and she just looked down and him and says "you know my rule, nobody is allowed to fall down the stairs in this house". I had to walk away before they all saw me laughing
My son, when he fell as a toddler, always looked to us to see if we reacted. I learned to say "TOUCHDOWN!" or "What are you doing on the floor?" or "While you're down there, can you see if there's any loose change under the couch?" If he was okay, he'd be distracted. If not, he'd cry immediately, and that's how we knew he was actually hurt.
^ How it should be done. If you freak out, they assume from your reaction that they've been grievously injured.
I've also heard of the trick of offering them candy as a "medicine," the idea being that if a slab of chocolate was enough, they weren't hurt, and if they continue screaming you might have to worry.
This necessitates walking around with uneaten chocolate, though.
I don't think I've ever heard of a carefree Republican.
If you want the analogy to be correct, you need the men side of your story to actively sabotage the stairs or push the child to make them fall.
Or the women trying to add or repair hand rails to make them safer, and the men saying "No that won't prevent falls." And actively prevent any repairs.