It's very clear IMO that whenever these right-wing extremist types talk about free speech, they're really talking about freedom from the potential consequences of their speech.
They insult people, then act the injured party when someone insults them back.
They spout racist shit on social media, with their workplace openly displayed in their profile, and then can't believe how that might cause co-workers and customers to believe they might get disparate and unfair treatment in the workplace. Or that their employer might not appreciate the negative publicity from that.
Reminds me of something I read a while back about an EMT (or similar) who posted on FB that they wouldn't treat black people. They literally told the whole world they wouldn't do the job that the taxpayers (which includes black people) were paying them to do, but it was "PC" and "woke" that got them fired..
"I didn't vote for Turmp, I don't even like him, but he's saying things that make sense to me." (Remember when right wingers tried that ploy? All the more hilarious as well, with Turmp's stroked out dementia stream of consciousness.)
Have you considered that the disappointment might be of your own doing? I mean, it's kinda silly to act like an asshole and then feign surprise and hurt when you get treated like an asshole. Just sayin. You sound like one of those types who thinks free speech means freedom from criticism and disagreement.
But anyway, I would never write up any one of my reports for something where there was no proof or evidence except for an anonymous email. I would investigate the matter thoroughly and independently, and perhaps maybe even ask the employee why someone might make such an accusation, but would take no action without proof (or confession) of some kind. Taking corrective action against an employee without that isn't just bad management, it's setting yourself up for trouble with that employee, should the accusation prove to be false.
Based on your attitude displayed here, and your proposed 'solution,' I strongly suspect your story is BS. Propaganda intended to silence 'woke' speech you disagree with by casting it in an unlikely negative light.