I had to do some stool collection and it took every ounce of willpower and a N95 mask to prevent me from vomiting everywhere. And that was my poop. I think it's more than cultural, there's a strong visceral reaction.
On the other hand, I can pickup my dog's poop no problem.
Nurses are heroes.
Similarly, it's probably useful for a primitive person to vomit on sight of a familiar person vomiting, collective protection. Definitely a trait to find out before going to space!
And yes, I do have a very vivid imagination.
Alternatively it could just be an overshoot of the behavior to recognize that you bleeding is a dangerous situation. These behaviors probably follow some gaussian distribution in their potential "effect" among the population and fainters are on a long tail of that distribution.
It is disgusting (I hated doing it) but you get somewhat used to it relatively quickly.
I think it's a question of exposure and tolerance, otherwise it'd be much harder for daycare workers, for instance.