Why is announcing pronouns a bad thing?

It's fine if someone chooses to put their pronouns on their signature. It's absolutely not fine if there is any pressure, coercion or suggestion that it's mandatory or desirable to use them.

It then becomes everyone sympathising with the distressed person and rushing to do this with absolutely no understanding of just what a slate of political items they're signalling.

It's just pronouns. Well, no it really isn't.

Yes, it's problematic because it assumes you're on board with the beliefs of gender ideology, which not everyone agrees with by any means.

It's like being asked to state what kind of vegetarian you are, or what kind of Christian, or what kind of Labour supporter. It presupposes that you are a follower of that belief system.

Exactly. I would say that you are unwilling to do so while sexism exists and send him some articles about sexism and stereotype threat.

Even leaving out the beliefs, requiring people to label themselves in the workplace by a protected characteristic is not appropriate. We don't require people to wear badges with their sexual orientation or their race, and for good reason.
This is no different.
It's also quite problematic as multiple surveys have shown that women are treated differently when their sex is known or assumed e.g. an email from 'Susan' may not be treated as seriously as an email from 'John'.
Requiring Susan to highlight her sex further by adding she/her further exacerbates existing gender inequality.

Who has their religion, or original ethnic, or political identities on a badge? These are private matters and have no place in a work setting.

Also, if you don’t believe you have a gender identity somehow floating invisibly in the ether separately from your sex, it is outrageous to be expected to pretend you do.

Even people who believe in souls don’t feel this is a public matter, much as it is important in the highest degree to them.

I suppose ‘gender identity’ belief could be seen like Phillip Pullman’s daemons in Northern Lights which sit on people’s shoulders. I’d rather have a badge for what I could decide mine is than have a gender identity one.

All people are at their core just humans with gender being irrelevant. Where actual sex does matter, ‘gender identity’ should not interfere or take precedent.

It’s a political statement which signifies allegiance to a side in a very heated debate regarding sex and gender, and it is not appropriate to include political beliefs in professional email signatures.

This is a very good one and shows the affect on women at work: www.huffingtonpost.co.uk/entry/what-happened-when-a-man-signed-work-emails-using-a-female-name-for-a-week_n_58c2ce53e4b054a0ea6a4066

www.google.co.uk/amp/s/www.independent.co.uk/news/business/news/gender-inequality-man-woman-switch-names-week-martin-schneider-nicky-knacks-pay-gap-a7622201.html%3famp

Gender identity is a belief. Many people do not share it. I cannot think of any other situation where using your work email signature to announce your personal beliefs would be deemed appropriate.

This is always a useful resource.

sex-matters.org/posts/updates/pronouns/

If I decided to state my pronouns are 'she/her', because I use pronouns relating to my sex not any concept of gender, someone then might determine that that means I am 'cis', which I don't identify as.

To avoid this I might specify 'they/them' but someone then might assume I identify as non-binary, which I don't.

It can seed a lot of assumptions.

fairplayforwomen.com/pronouns/

Ooops - here's the archived version: archive.md/zdtxP

www.wsj.com/articles/asked-your-pronouns-dont-answer-lgbtqia-sogie-gender-identity-nonbinary-transgender-trans-rights-sexism-misogyny-feminism-11643992762

yogyakartaprinciples.org/relating-to-the-right-to-privacy-principle-6/

G) Ensure that requirements for individuals to provide information on their sex or gender are relevant, reasonable and necessary as required by the law for a legitimate purpose in the circumstances where it is sought, and that such requirements respect all persons’ right to self-determination of gender;

H) Ensure that changes of the name or gender marker, as long as the latter exists, is not disclosed without the prior, free, and informed consent of the person concerned, unless ordered by a court.

Basically, they could be forcing someone to 'out' their true gender or force them to decide on a gender when they are not ready to.
There are also plenty of Data Protection issues as to who would then hold data on someone's internal gender and why, how that might be updated if someone's gender changes or could be used against them.

It's not a great idea to uniformly demand this, but obviously people are free to specify their own pronouns if they are happy with the risks.