Well, according to Nordic tradition, Kyne is a warrior at heart, Warrior-Wife and widow to Shor to be specific. She is unlikely to be squeamish when it comes to violence and, like any red-blooded Nord lass, she'd probably see death in honourable combat as a good way to die, whether the dead be man, mer, moose or mammoth.
As you pointed out, Kyne's Sacred Trials has you tracking down and killing a variety of animal guardian spirits (presumably without eating their ectoplasmic meat or tanning their spectral hide!) and Froki Whetted-Blade states that, "Kyne teaches us to respect the beasts and blesses those who will face their champions.".
If he is to believed, I imagine that Kyne would not condemn hunting for sport. She might commend hunters who seek glory in killing great beasts. But, the hunter must respect the animal, likely give it a fighting chance and honour it with a swift death.
I don't think that Kyne would, as legend has it, be one of the first divines to agree to the creation of the Mundus and remain committed to birthing it, if she was opposed to the notion mortal beings killing each other.
We can't see the big picture. Maybe she's happy to see her creations die because it brings them closer to her in death or something? That or they maybe provide game for hunters in Sovngarde...