Does partisan activity on social media hurt the military profession?

Overall I'd say yes. While we are prohibited from partisan activity when on active duty, the average civilian social media user has no idea of that fact. So once we are out of uniform and can express our partisan beliefs those civilians who know one has a military background believe what is posted, as you said, are a representation of the entire force.

There is an absolute prohibition on military personnel engaging in partisan politics. However, I am no longer serving. I am a "PFC". I believe, however, it is my duty to speak for those who cannot speak for themselves, especially when other veterans speak out with opinions that I believe are harmful to America, it's citizens, and those who now serve. I also do my best to never give anyone a negative impression of those who serve/served, but I see no merit in the argument that keeping silent is the best manner in which to accomplish that goal. Always keep in mind that silence is reasonably assumed to be acquiescence.

How is having a partisan conversation on social media any different than one in person? I realize there is more of a measure of anonymity, but that's beside the point. All members of the military are allowed to voice their political perspectives, and like any other American, should never be denied that right. We have our rules on how to do so, both in and out of uniform and during/after service, and then there are levels of respect that should be maintained, regardless of one's current status. Not just because we're Veterans, but because this is how humans should act in general. There is nothing wrong with pushing a partisan perspective and having the ensuing debates. I welcome those discussions, so long as they are civil and respectful, and there are some people, even on RP, who are capable of having them. Those which are not, I block the individual and move on, while pointing out where they went wrong with me. As for those who break the rules (such as the TOS here or the UCMJ in general) that's on them.