First of all we should define empathy.
If you mean cognitive empathy, i.e. knowing/understanding what the other person is experiencing: yes
This can definitely be learned by careful observation. Psychopaths train themselves to read others. People with Asperbergs can learn it, as far as I know.
If you mean affective empathy, i.e. having a (socially) appropriate emotional reaction to another person's emotions, sharing their emotions: probably not
While I generally agree with your brought points, I think I do disagree a bit here. I think there are people who are emotionally cut off from themselves (often losing a good deal of bodily awareness) because of emotional trauma or ongoing emotional stress. Such people are perfectly capable of being emotionally aware and practicing affective empathy, but cannot consciously access that level of emotional awareness. Such people can, I think learn affective empathy.
Plus, I think it likely that even those with a limited amount of affective empathy can learn to attend to it more and learn to see it as a valuable resource.
I think the latter has a lot to do with mirror neurons. Not sure how much of a fixture their strength is, i.e. if you are born with very active mirror centers or if they develope as they are trained.
My hunch is that a good part of it is either hardwired or set at an early age.
I suspect it's like most things... how good you are at something depends upon inherent skill, training, practice and other life experiences (external feedback, critical growth periods).
It's funny you should mention this as I have long been wondering about the relationship not only between affective empathy and mirror neurons but also between those two and Fe/Fi
And what would you say associates with what?
F7 vs T5
I think it's interesting that as for as the neocortex goes, there are two "mirror neuron" regions (using Nardi's terms):
F7 - "Imaginative Mimic" - Mirroring other's behavior, imaginatively entering into other situations and "what if" scenarios
T5 - "Sensitive Mediator" - Noticing and responding to social cues, wondering how others evaluate you, being embarrassed
I think F7 sounds kind of Ne (with a little F/Fi flavor, as far as mirroring), and T5 sounds somewhat Fe. Nardi did find that FJ was associated with T5, and Ne with F7. But the T5 association was a little more complex (see below).
Regardless, I do think it's interesting that mirror neurons are divided into who separate regions, both of which might relate to empathy in different ways.
Nardi and T5 (an aside)
T5, the region of the neocortex related to attending to social feedback and feeling ashamed/embarrassed, is used heavily by Fe types (and seems to be an obvious fit for at least some qualities we associate with Fe).
Conversely, ISTPs tended to show the least activity in T5, with some ISTPs never showing any activity in the region no matter how embarrassing the situation in the lab. INTPs generally showed low activity in T5 as well. However, when that region finally activated for INTPs in the lab, it tended to spill over into neighboring speech and movement centers (which may be one reason why most of us, regardless of type, get clumsy and tongue-tied when embarrassed).
T5 is also used by heavily by male FPs, but far less so by female FPs... but what does that mean (Nardi didn't speculate on that, that I recall)? Could it mean that male FPs tend to feel more socially constrained than female FPs (and so perhaps are less likely to be visibly quirky)? Do male FPs learn to blend socially for some reason (social pressure)?
Of course, some studies have also shown that men tend to use T5 for facial recognition, which might explain a greater use of T5 in male FPs.