I have read somewhere that losing a limb naturally (e.g. tearing, biting) stretches and closes up blood vessels in a way that cutting with a scalpel doesn't, and that natural limb loss has less bleeding than surgery/stabbing/Hollywood suggests.

I have no idea of the truth of this, but there are anecdotes of farmers losing limbs and walking back to their tractors, driving back to the farm, then driving themselves to hospital, and the like.

Yes that's kind of (but not entirely) my point. It's actually quite difficult (for a human) to lose a limb in a sudden accident without immediately dying. So it's exceedingly rare (~all historical civilisations) to be injured in such a manner, it's at least somewhat rare to survive such an injury (historically just about impossible), and even if you had flawless regeneration you'd be at a serious physical disadvantage (ie from an evolutionary fitness perspective) in multiple ways for an extended period. Meanwhile carrying said trait is consistently killing people off the entire time.

Yes, that was my point. Amputations are quite uncommon, they usually lead to death (blood loss / infection, starvation, vulnerability to predators, etc) For relatively intelligent species they are even more avoidable. Regeneration would add very little benefit from natural selection point of view. Most body parts are durable enough for survival and reproduction.