Of course a doll is a doll and not a human. A spider is a spider and not a baby (so probably less capacity- but not none- for feeling pain).
But since young children are seldom left in charge of babies and in a position to do them serious harm, dolls and insects are practising tools for them to learn empathy.
And then as they grow older to learn the rules that surround killing:
you may kill when there is need, but only certain species or under certain circumstances
when you do kill you must do so as quickly as humanely as you can
inflicting pain for the sake of it is never right
I grew up on an island where we needed fish I caught (from a very young age) to supplement our diet. But I knew the rules: why the fish was killed, when fish could be killed (never fish under a certain length or certain fish at a certain time of year), and I knew how to do it humanely. And for those reasons, I would never have transferred that to pulling legs off a spider.
Again, I swatted mosquitoes- for understandable reasons. But it would never have occurred to me to burn a mosquito (though I had access to matches from quite a young age) because that would be needless infliction of pain.
I would see this as a teaching opportunity rather than "there must be something wrong with this child".