Rabbit snare methods

I thought I remembered from my Boy Scout days (many moons ago) that you snared rabbits with a bent over sapling and a figure 4 arrangement with string.  

But YouTube seems to be saying everyone uses wire snares that are just anchored to a branch.  With a stiff wire snare, how does the rabbit die?  It seems like it would have to keep pulling on the noose to strangle itself.  Whereas with a springy sapling it seems like it would die faster.

What am I missing about how a wire snare works?

It needs to include some way for the snare to tighten and hold the tension. The animal pulling on it cinches it ever tighter. I remember reading that you could use a bent penny with some holes drilled into it to act as the tensioner, but I never really figured out how it works. Snares are not a legal way to hunt here, likely something about them catching too many of the neighbors cats, I would guess.

I suspect that a break-action air rifle and a couple of tins of ammo would also be enough to rest assured that you could eat all the small critters in your neighborhood if it came to that. You can get a decent one for about 100 bucks, and it is a great way to practice shooting without all the noise (and cheap ammo to boot!).

A simple snare lock can be made with a penny or any piece of angled metal roughly 3/8 in on a side. Two holes one on each leg, just large enough to allow the snare wire to slide. Tension on the wire allows the wire to slide tight but for it to loosen you need to cock the lock to allow it to slide open. There is not always a convenient spring pole.  A neighbor let all his rabbits loose when he got tired of managing them. I was catching 3-5 kits at the same time in one live trap, we were over run. I finally resorted to setting snares to reduce the population that was destroying the garden and new trees. Piano wire snares without locks worked well, the sliding wrap a little long worked as a lock. Snaring is a good skill to learn, with any trapping you need to be responsible and maintain the line. Harvesting animals this way is not for everyone.

I think I'd rig up an old fishing pole as my spring pole...  Is there a reason why 20# monofilament line wouldn't work with a spring pole?

No reason the monofilament wouldn't work. Being creative and open to expedient ideas makes a difference. We've used ATV flags as spring poles no reason robust fishing poles wouldn't work. Fishing might be easier than snaring in some places. Whatever brings in the calories.

I heard that snares often capture and kill cats, unintentionally. I was wondering, how can you know what kind of animal will get caught?
I'm not against hunting etc, just curious if you know more about these issues?

In the UK lethal snares are illegal. They have to be designed to slide freely and not tighten to kill or injure. Just hold them securely.

The idea is that the trapper checks them frequently and either releases them or dispatches them. It prevents the wrong species being harmed, and doesn't cause slow suffering.

There is specific legislation on how snares are constructed, including minimum wire thickness, locking mechanisms and such.

One of the problems with trapping is that it can be indiscriminate. Not a problem if it is a survival issue. However, the use of leg stops, and intelligent placement is prudent. I don't think I'd be setting snares in a suburban area. I have destroyed illegal bear snares in the woods around here. Snares are not the most humane or legal way of trapping in some instances.