The Kness Catch-All mouse traps are made of metal, so no one chews through those. The smaller traps do, indeed, live trap the mice. Some mice are killed by the larger traps; I've read that their necks get broken by the mechanism. There's a little window so you can see if there are mice inside, and to release the critters, you lift the lid and left them go. Or not.
These traps don't require bait; if you place them along a wall where the mice naturally run, they seem attracted to the entry hole. I originally baited them, which only resulted in a messy trap that needed to be cleaned. I have these in my coops and in my house.
BTW, the Kness factory is directly across from the property where I grew up in southern Iowa. It is family-owned, and I went to school with some of the -- at that time -- kids. At least at one time, Kness traps were used in the White House.