I would argue that it's a disadvantage only from the self-centered perspective that modern society encourages us to cultivate. Empathy is an incredibly useful tool for humans that exist as part of a community, but our societies are structured to benefit those who seek to maximally exploit others, our role models are sociopaths, and the cultural narrative is that our only value is either in being them or being useful to them.
It can be an advantage when it's been tuned and trained properly. When it hasn't been, it mostly just leads to additional pain for the empathetic person, and eventual withdrawal.
Parents need to be taught how to recognize children that are highly empathetic, and given the tools to teach them to train and deal with that aspect of themselves. Otherwise they'll experience great emotional pain in their lives without any of the tools they need to cope with it. That can have a rebound effect, leading to emotional numbing and social withdrawal.
Empathy is great for the society, a burden for the individual. Very empathetic people have a hard time minding their own path in life for they get caught in the suffering of others. Regardless of setbacks and disilussionments, empathetic people feel good when helping others.
This chain of comments reminds of what I read in a paper regarding multilevel group selection, namely:
"Selfishness beats altruism within groups. Altruistic groups beat selfish groups". Though there is a bit of controversy around that if I remember correctly.
