Okay, a couple of things. First I respect your dislike of trapping, it's definitely not "fun" for the animal and I can understand your dislike of it. I'll address your points in turn.

First, you could have used "foot-hold traps" as that is an accurate term for these types of traps. The word "bear trap" does not simply bring to mind a foot-hold, it brings to mind a very large trap often with "teeth", which is used very infrequently in modern trapping (at least in U.S. which is where I'm going off of).

Second, no trap is designed to incapacitate by injuring an animal. Any life trap (like the foot-hold) is designed to hold the animal without injury. If a foot-hold actually injures an animal, it's because the trapper used the incorrect trap and is very much looked down upon in a trapping community. Other traps, such as "conibears" are designed to kill immediately. These are generally used on beaver, muskrat, and less commonly on land in "box traps", but most trappers I know of don't like using them because of the possibility of getting incidental catch or pets.

Third, It can take time to check traps, but there are regulations to keep people checking traplines quickly. Any live traps must be check once a day. Kill traps have to be checked once every four days. So any animal that is caught in trap that does not kill it immediately can only be in there a maximum of 48 hours and that's extremely unlikely. Most trappers check their traps in the early morning and animals are usually caught at night, so most of the time, animals are in the traps less than 6-8 hours.

"Have-a-Heart" style cage traps are available, but are extremely unwieldy and not practical for a full trap-line, not to mention have very low catch rates due to their conspicuous nature. Trappers don't avoid these traps because they're "too humane" or anything, they are just not practical.

Finally, almost all animals that are trapped are done so because they are in abundance and need population control. Animals like muskrats, beaver, raccoon, and even bear are nuisance animals because we have eliminated both their predators and habitat needed to keep them in a stable population. For many of these animals, hunting is either not enough or not practical to keep their populations in check, so trapping is used for these purposes. Now, I'm not saying that trappers are only doing it for the environment, most enjoy the outdoors and find it enjoyable to get out there in the early morning when it's quiet and this is a hobby that, if done skillfully (there is a lot of skill involved in trapping animals), it can either pay for itself or make the trapper a small profit. In addition, the fur obtained by wild trapping is still more humane (in my opinion) than fur obtained from a fur farm, and this activity will supply much of the demand that is out there for fur. Either way, trapping does serve a real conservation purpose, it's not just needless killing.

I know that was a lot to read through and I understand if I don't change your mind on this matter, but I hope you take the time to read through it and, at the very least, try to amend some of the misconceptions that are often believed about trapping.