I believe you are confusing empathy with sympathy. These are often confused but they are not the same thing.
Empathy is simply the ability to understand and imagine yourself in the world from the emotional and psychological point of view of someone else. It does not mean you have to sympathize with them, their words or their actions or even that you have to like them.
Empathy is simply a path to a better understanding of other people’s motivations, drivers, and point of view. Being empathetic says nothing about what you do with that knowledge and it implies no moral position.
While yes empathy will often lead to sympathy (because of this better understanding) it is possible to empathize with someone while detesting everything they represent. In fact, that is one of the most valuable uses of empathy, understanding what motivates those that are your most counter opposite. In this regard in particular, empathy and the ability to empathize is ultimately a self serving skill.
And as it is a skill, anyone can learn it. You may not be able to explain or teach someone how to experience empathy but there are people can, people who specialize in exactly this kind of personal development. From pastors and priests to therapists and psychologists there are a wide range of options available to people that want to improve their ability to empathize.