As usual, I'm high, so excuse the garbled half finished thoughts
Empathy as I see it, is a beings ability to experience an emotional state as if they were the entity they are empathizing with.
Empathy is not the perception of that state in another, it is the experience of actually feeling an emotion as if one were that other being. The information that triggers the emotion can come through any of the functions, one can pick it up through feelings, one can sense it through body language or other physical aspects, one can come to a logical conclusion that the other is experiencing an emotion, as well as intuit their way there by putting themselves into the others "shoes" or whatnot. While the feeling function may be about perceiving and manipulating "emotions" and "feelings" in others, all of the functions are able to trigger emotional responses within a person.
When it comes to feelers, they may or may not be more comfortable, immersed, or secure in these emotional responses. It's a matter of the available emotional experiences the person has to draw on. An NT who has personal experience with love and affection will be able to experience them empathically in a truer sense that a feeler with limited or no experience who resorts to synthesising them.
I had intended to say more, along with stories of empathising with drowning bugs in my pool, but I've lost the thoughts for now. heh. maybe I'll come back to it later. Just before anyone jumps on the bug thing, it is NOT sympathy I'm talking about. I FEEL the despair and panic as if I were that bee frantically twitching to no avail, not a feeling of sorrow or worry for the bugs condition. I also have stories of empathy for killers, abusers, and other darkness. That's another post though...