I think what sounds so jarring about 'empath' is that it is used in the way we typically use job titles or vocations, not the way we usually describe personal attributes.
'I am a' phrases might include: [job], parent, woman, pensioner, keen musician, athlete, gardener. Those are all externally apparent or verifiable facts about what we are and do.
We don't use 'I am a' phrases about personal qualities or skills, that while real, are often more subjective, like intelligence, sensitivity, generosity, kindness.
We might say 'I am a very empathetic person', or ...intelligent / sensitive / generous / kind person. What we are though, is a person. The degree to which we are any of those qualities is inevitably varied, subjective and comes to the fore more in some situations than others.
Whereas saying 'I am an empath' sounds as though there is some objective standard of empathy that has been achieved, rather like 'I am a chartered surveyor' but more so, that this is a perpetual, immutable feature of that person, like 'I am a white woman' but, even more so, one that is always to the fore.
It sounds very odd to my ear, to hear a personal quality, that exists among many others within that person, one hopes, described so monolithically.
Whereas, used to describe a race of aliens, the grammar of 'an empath' makes perfect sense; one either is, or one isn't.