The pain of having your skin pierced is nothing compared to the pain of having a major organ ruptured or a broken bone.
On the contrary, many people have said that getting a paper cut is worse than a broken bone, unless the bone itself has shards poking through the muscle and the skin where the pain receptors are. Some people don't even notice they have a broken bone because they don't move it enough to pierce flesh. My brother said his broken wrist was only "mild soreness", with no sharp pains, so he didn't even know he'd broken it. He thought he'd only bruised it.
(As for organs, some do have pain receptors, some don't, and sometimes things lead to organ pain in other ways, like muscle cramps.. etc.)
Edit: Oh, and I broke my collar bone when I was a kid and had no idea and it fused back together a little wonky. I always forget about that. lol
In fact, look up the lovely little practice of Scarification (also tatoos, on a lower level) for examples of people very, very repeatedly cutting their skin open (and then potentially stuffing something into the wound), and not passing out from the pain.
I'm aware of practices such as that, and I already addressed that sort of pain. You can withstand pain with a determined personalty and/or training, such as going into a parlor for the purpose of getting a tattoo or performing a ritual of scarification, which is far different than being thrust into combat or dealing with a pain you weren't expecting. Those people are going into it, knowing what they are getting into and are fully prepared for it, and the area to be scarred/tattooed is one small area that the brain can easily redirect focus from. Also the people performing the tattoo/ritual on the recipient are usually talking to them a lot to help them keep their mind off the pain.
But this goes back to why pain exists in the first place - it's there as a means of conveying information to the creature. Creatures are built to withstand the level of pain they can reasonably expect to encouter. If an amount of pain inflicted by stepping on a sharp pebble is allowed to add up by every sharp pebble a creature like an elephant would be stepping on on a daily basis, they would constantly be fainting from the pain of walking around.
You might talk about "training", but there's a certain level of conditioning involved. You mentioned something about "larger creatures don't have larger mental capacities", but that's beside the point - having more intelligence doesn't make you less sensitive to pain by any stretch of the imagination. Being sensitive to pain requires MORE mental development - you have to specifically develop the nerve sensors to convey that information to the brain. It's very easy evolutionarily to just dampen the pain if it becomes necessary because getting a few mosquito bites causes creatures to pass out from the pain.
Likewise, if elephants could be killed by relatively tiny creatures performing a few minor nibble attacks on it, every tiny creature in the same environment would have evolved to eating elephants, because they're such wusses that a shrew could kill an elephant by forcing it to pass out after a few nibbles, and then they could go in and eat the elephant's eyes or other tender flesh.
In real life, elephants often wind up pretty battle-scarred from fighting off larger predators like lions. (Who have four claws to attack with in one swipe - that's four separate scratches! Four pain points in one swipe! It only takes six of those to make the elephant pass out!)
Animals are naturally resistant to pain just because they simply have to be, or they wouldn't survive. It's only humans who get all sensitive about silly things like paper cuts. It's just people who have forgotten how painful it was when they got major bone fractures who compare their most recent paper cut to their bone fractures and say the paper cut was worse. (That forgetting how painful it was is a natural coping mechanism - as is just plain not fainting from pain.)
Further, as someone who has had paper cuts and needle pricks and tearing out major ligaments and broken bones, things are more painful when you don't have the adrenaline pumping for one thing. A paper cut is painful only because you focus on it. I've had cuts I didn't even notice until someone pointed out to me that I was bleeding. If I had thirty cuts I didn't even notice I was bleeding, I wouldn't be fainting from the pain I wasn't feeling. When you're being swarmed by angry badgers, you're probably thinking more about keeping the angry badgers away from your eyes than about how much five different cuts hurt.
Besides, remember that old joke? The one where a guy complains about how his foot hurts, so the other guy breaks his arm. The first guy screams in pain and asks why he did that, and the second guy responds, "Well, now you're not thinking about how much your foot hurts, are you?"
An insignificant wound just isn't painful unless you're focusing on it. (And that's how those scarification people pull through those procedures - they just don't focus on it.)
All in all, I was enjoying this friendly debate, but I get the impression that you aren't really considering all my points, so I think I'll bow out now.
I'm not sure what you mean by that...
I'm responding to your points in order. I'm just disagreeing with the premise of your argument.