Isn't it a bit hypocritical to object to the phrase "culture war" on behalf of a set of progressive ideas that included the idea that speech is violence?
There have been metaphorical wars on lots of things (drugs, cancer, etc.), but if one blurs the line between physical violence and violence inflicted verbally or conceptually, it's hardly fair to turn around and complain, "hey, no, it's not a war when we call you racist because you say things like that sound like 'merit' when we say things that sound like 'structural inequity'!"
Secondly, the battleground is littered with civilian casualties. Oh, wait, no, they're all combatants: because if you're not an anti-racist (with accepted ideas about how to be anti-racist only), you're a racist. This doesn't help to undermine the analogy to war.
Thirdly, there is an extremely unfortunate tendency to elevate disagreements about the best way to balance competing concerns into the idea that any disagreement indicates that "women’s rights and LGBT rights are negotiable" as a whole and not just an issue where different rights conflict. In addition to being warlike, this makes it very difficult to articulate the difference between really fundamental needs like access to essential heathcare and luxuries like additional entertainment opportunities--which makes it much harder to mount a concerted defense against the most radical right-wing attempts to deny fundamental needs because those people really don't seem to think women, etc. deserve rights.
So...I agree that the term is unfortunate, but the progressive left shot itself in the foot on this one. This war was their doing: they started it, and the main contribution of the radical right was to go, "Oh boy! A war! We can do a war!"
Not sure how to roll things back to a place where we can calmly have discussions about how to ensure people have the rights that they deserve. Saying it's all the right's fault doesn't seem like a promising strategy to me, however.