I do not think that using sympathizer to refer to anything political or societal in nature, or to a group or unspecified individual, would come across as synonymous with supporter in a genuine way. I
I could maybe see it being used to refer to an interpersonal relationship, one in which genuine sympathy is expressed, rather than ideological approval. In the political and anti political circles in which I engage, sympathizer is always used negatively, synonymous with and often vaguely referencing nazi sympathizers, such as fascist or misogynist sympathizers. People who maintain relationships with a called out abuser, do not confront them, or in the case of celebrities, still enjoy or promote their work, are often called abuse sympathizers or sympathizing with [the abuser(s)].
The only other way it is used is sarcastically, like an inside joke, since it is such a common word in the culture and not used as much outside of it. For example - “not to be a winter sympathizer, but I love snow”, implying that winter is a bad thing that would be not acceptable to the group to speak positively of or support. And that is a
A difference between support and sympathizer, I think, is that sympathizers don’t have to do as much, maybe just didn’t delete someone on Facebook, maybe just didn’t do enough to speak out against someone. Often it is speculative thus safer than calling the person a supporter, as being outed as one can damage ones place in the group. A sympathizer doesn’t have to do anything, they can have not done enough. Often they are entirely unaware of the situation, but of course that is when the word is used inappropriately.
Sympathy, rather than empathy, implies negativity. Interpersonally, if I express sympathy towards somebody, it is because of something difficult they are going through (sympathy cards in the greeting card aisles are given to people experiencing illness, death, etc.) In this case the negativity is not a tragedy or difficult time, but a bad ideology or practice.