That seems to me to be a gross overgeneralization (not to mention that it's an obviously self-serving one). Groups that have historically been defenders of free speech and related first amendment rights are generally left-leaning or bi-partisan. The ACLU comes to mind, though they have been known to waver under sufficient political pressure. Probably the staunchest group I know of currently is F.I.R.E., one of the few I contribute money to support. And they are pointedly bipartisan.

I think a more useful idea than yours is to look at multiple organizations and individuals, regardless of political affiliation/orientation, and evaluate where they stand and how they've performed in practice under pressure on free-speech issues. To the extent that a group or individual has stood up to protect free speech rights for those they personally disagree with, the higher they should be ranked as serious supporters of free speech rights.

Going the other way (determining political orientation and then judging absent specific evidence which way a group or person will fall on free-speech questions) is counterproductive. And it misses the fact that anyone can claim to be "left-wing" or the opposite, but that doesn't really tell us all that much about his/her commitment to the first amendment or anything else. In this era of political correctness and wokeness, I'm constantly being told that I'm obviously: a fascist, Trump-supporter, Republican, reactionary, etc., as well as an "obvious" communist, libtard, ad nauseam. Apparently, my opinion on my own political views isn't relevant. But if anyone cares, I see myself as a socialist and liberal ironist (see Richard Rorty's CONTINGENCY, IRONY, AND SOLIDARITY). I'm fanatically in favor of free speech and have been since before I was a teenager.