klugeryid,
“We both agree, we’re just haggling over price.
So now the question is what’s worse
A child playing with matches or a child being מחלל שבת?”
Nope. I agree with WinnieThePooh, but couldn’t disagree with you more.
The use of physical force in each scenario WinnieThePooh described is intended to stop the dangerous act as quickly as possible, not to provide chinuch. Smacking the hand away is the quickest way to prevent it from getting burned on the stove. Yanking the kid away from the street is intended purely for protection, and physically separating fighting siblings is to prevent injury.