But we don't know exactly what parts of the brain are responsible for the experience of empathy! Claiming that "mirror neurons" are the source of the psychological experience of empathy is crude reductionism that almost all neurophysiological researchers reject. A complex psychological state is not produced by a specific set of neurons firing. Example: recall of personal memories. Early researchers assumed there was a "memory cortex" where all episodic memories were stored, and specific memories would be elicit when specific neurons were activated. Alas, things are much more complex and interesting: recall of a specific memory can involve neural activation in completely different parts of the brain. There is no specific set of neurons "responsible" for expressing a memory. In the same way, researchers have not discovered any specific area of the brain "responsible" for the psychological experience of empathy. In the same person at different times, the experience of empathy involves neural activation in different parts of the brain. Thus it's questionable to claim that the parts of the brain associated with empathy are evolutionarily "early".
[WS:] Based on what I read, MRI based techniques allow linking emotions to particular regions of the brain. This is not my area of expertise, but if true, this would weaken, if not contradict the above claim.
However, I also listed another argument that is independent of MRI mapping and instead depends on the stages in human development. Empathy occurs in children relatively early, well before higher cognitive functions develop. This suggests that it is associated with more "primitive" from the evolutionary standpoint brain regions, shared by many species. If fear or joy are emotions shared by humans and animals alike and appear early in human development - it is likely that empathy (which also appears early) is linked to the same brain functions shared by human and animals.
As to your point on morality and human exceptionalism - I did not mean to imply that you subscribe to it, but that it is well ingrained in our stock knowledge, and defines the frame of thought even of otherwise critically thinking people. If you abandon that frame of thought, the whole discourse changes quite radically - it is the differences between humans and animals that would need to be proven instead of the similarities (or the "null hypothesis.")
Wojtek