The phrase "insensitive to suffering" might mean--(1) "culpably unaware of it" or "unsympathetic to it." Or it might mean (2) "trivializing it" or "giving it too little weight."
If you shoot a rabbit, that's got to cause the rabbit to suffer quite a lot. Surely there's a good chance of not achieving a "clean kill." If you think the suffering you cause is worth it, for the pleasure of eating rabbit stew, then arguably you are trivializing the animal's suffering. So you are insensitive to it in the second sense.
It could be, but doesn't have to be, that you are also insensitive in the first sense. In fact, hunters I speak to (I teach an animal rights class in Texas) often seem very invested in the notion that it doesn't hurt animals much to be shot. They don't respond with sympathy to animals that surely are, in fact, suffering.
Even humanely raised beef cattle do suffer--when they are branded, castrated, dehorned, and probably when they are slaughtered. I don't think we can dismiss the worry that people involved in the practice, whether directly or as customers, are insensitive to suffering in one or both of the two senses.