I disagree with this. This is particularly relevant to me as I work in the civil service, which is supposed to be politically neutral, but it applies anywhere.
Those who choose to put their pronouns in their signature are making a definitive political statement.
Not putting pronouns in your signature is not an equal and opposite political statement. It’s impossible to say whether those who don’t put their pronouns in are doing so in protest or are merely indifferent.
Several people I work alongside have pronouns in their signatures. They include the head of my division. Because he has his pronouns in his bio, to be equal with him (and others) I should be allowed to state in my signature that I disagree with the practice.
Should I choose to write something professional in my signature, such as “I consider all pronouns should be sex based” it makes it obvious that what we have is a competing situation of rights.
I would argue that stating your pronouns (when they are perfectly obvious) is a political statement and not a useful exercise to help people identify you correctly. A good lawyer should be able to argue such a case, I believe.
This case is unhelpful as this aspect was not explored. It may be that his signature was noted as being unprofessional, and therefore a sacking offence. It appears however that some hyperbolic statement was used about harming people, which a decent lawyer could also have pulled apart.
So I disagree that pronouns or omission is an equal position. I would very much like to see this case properly argued, or preferably a better case, where there was no question of unprofessional conduct, beyond the equally unprofessional conduct (in my opinion) of including your political position on pronouns in your signature by listing them.