I was taught early on that you don't use deadfalls for rabbit for just that reason. Some animals are more susceptible to crushing damage and targeting them with deadfalls should be avoided. Squirrels (tree and ground), ground hogs and raccoons are some of the best animals to target because they are not likely to have meat spoilage from the crushing. It is important to size your weight properly so that it will kill the target species but not crush it to the point of rupturing internals. I usually go with a 5:1 weight to target species ratio. I have not had an internal rupture in the last few dozens of deadfall kills. I saw it occur a couple years ago at a primitive skills meet but the guy who did it was corrected IMMEDIATELY as wasting an animal is a very severe taboo in that community.
Targeting- It has come up in this thread that people are concerned about bi-catch (killing unintended species) from deadfalls. Just as snares and coni or grasping traps it is not hard to target only the intended species. There is always the chance that bi-catch could occur but knowing where to put the trap and how to bait it and the proper trigger tension will prevent it most of the time. I have seen a couple cases of bi-catch over the years in small game deadfalls but that was because of traps set in improper locations. To learn these skills properly then you must go far beyond the mechanical construction of the trap alone.
Use of Animal- Just to be clear on what is done with the animals trapped that I post on Paleoplanet and occasionally here on WSS, they are all used to the fullest of my/our ability. Squirrels for example will have their tails removed and split for use as string silencers, hides are always tanned and used for handle wraps and other projects, meat is obviously eaten (most often wrapped in bacon and roasted over communal camp fire), left over bits with fur "limbs, mask, etc..." are given to any members who tie flies for fishing and usually dyed different colors with natural dyes. The animals which lose their life are respected and given proper treatment, nothing is wasted. Larger animals provide more parts like bones and sinews from deer and teeth from nutria and beaver which are used for chisels and wood shaping tools. This has MUCH less environmental impact than buying everything at the grocery store. I made a reed grass basket last year that was roughly the size of one sold at target that has a "made in Indonesia" sticker on it. That's a heck of a carbon footprint for a simple basket. Meat and wild edibles are the same way. Land used for farmed meat is far less useful to the natural environment than wild lands that have animals removed for food and tools. Its all about prospective and what your goal is. Killing any animal is a serious act so it should only be done when there is a use for that animal.
P.S. Don't take that last part as being against farming for meat, I love my T-bone steak as much as the next guy!