In many larger organizations these days, almost everyone includes their pronouns in their email signature except, of course, those who ("against the rules") do not even have an email signature!
But aside from that, I've noticed that almost no one ever mentions pronouns. A few do list them with their videoconferencing "name", but not too many. But for when introducing oneself at regular meetings, I never hear anyone say things like, "I'm Rob. My pronouns are he/him/they. I'm a communications specialist..." The exception to this, logically, is events and webinars related to diversity and inclusion, where the presenters frequently include their pronouns and invite attendees to share a bit about themselves, including pronouns if they wish, in the chat.
Here is my broad perspective on this: Once an organization allows pronouns in the email signature, most people will add them. Some will be keen on doing so, others will add them because it's somewhat expected. But that's where it ends.
"One and done."