If your kid wanted to go outside without his or her coat on, and it was really cold, would you let them?

I would. The ding dongs!

When I had a daycare the kids would usually walk to school, which was around the corner. One kid didn’t want to wear his coat. He was trying to push buttons, and it probably would have worked with his Mom, but not with me. I said, “Fine, but you do have to take it with you.”
“OK, but I’m not going to wear it!” he claimed.
“I don’t care,” I said.
I watched him. By the time he’d gotten to the next-door neighbor’s house he’d put his coat on.

Ding dong.