> why does employer not say "dear whoever, this employee is clearly an idiot, and whatever they say on their private twitter does not represent our views"

Because:

1. It wasn't private, it was public and tagged the subject of the abuse directly.

2. "This employee" was on company time and actively representing the company at the time.

It doesn't matter what role you are, you are always at risk of getting fired if you embarrass your employer.

Why would you want to keep toxic person in a company?

Then they should be fired for being toxic, not for stepping on the wrong guys toes.

I think that's exactly what happened.