1. Everyone cares more about free speech for opinions they themselves hold. So a defense of free speech might be merely opportunistic. This is the same for the left and the right. So a free-speech issue involving a faculty's right to insult a right-wing group like the NRA will rally left-wing professors to it. An affront to free speech that prohibits certain kind of conservative speech will arouse the ire of the right-wing outrage machine. For example, suppose political opposition to affirmative action might count as a "micro-aggression." We only know if someone is really devoted to free speech per se if sh/e defends the right to express distasteful opinions.
2. A free-speech issue only arises with an unpopular opinion. In some sense, then, the fact that the right has more issues might have to do with the fact that certain conservative opinions have become unpopular. This is a good thing, right? I'm not saying that nobody is racist, but racism is publicly unpopular, in the sense that almost nobody wants to be perceived as a racist anymore.
3. So the right "seems to care" more about free speech in the academic context, because academia skews left. Most of the left-skewing of administrators is simply a response to legalistic matters and public opinion. If they were really left-wing the world would be a much different place.
4. Just because the right has an outrage machine designed to magnify any perceived threat to free speech or the censuring of conservative opinion does not mean that these threats are not real.