First, I use them. You use them. We all use them.
What people really mean when they say they won't use "pronouns" is that they refuse to self-identify their pronouns.
But pronoun identification is not a burden, or something that will only serve one small segment of the population. It's a curb cut.
Curb cuts are the small ramps cut into sidewalks at crosswalks. They were mandated by law through the ADA to assist those with physical disabilities and mobility difficulties. Yes, they help people who use wheelchairs, walkers, or other mobility assistance. But curb cuts also help people pushing strollers, pulling wagons, making deliveries, pulling wheeled luggage, riding scooters, etc. Anyone who's had to push, pull, or ride something across the street from sidewalk to sidewalk has had their life made easier by a curb cut. (If you're interested in this idea, check out the work of Angela Glover Blackwell who has written more on the Curb Cut Effect.)
Regularly self-identifying pronouns is also a curb cut. Because yes, it will help those in the LGBTQ+ community.
But do you know who else will benefit from pronoun identification? People named "Alex." Or "Jamie." Or "Adrian." Or those with any other first names that society considers to be gender neutral.
Anyone with a name that does not have gender connotation commonly recognized by mainstream American society will be helped. This includes people of color, immigrants, people with very modern names, people with very traditional names, and many people I've neglected to mention here (because I know I have my own gaps).
I'll give you one big example of how pronoun identification would help me in my own day job. Part of my job involves calling or sending a letter to people who have submitted an application for a service. If the person's first name is one that I'm not familiar with as gendered, I have occasionally found myself embarrassed by misgendering, or awkwardly using "they" to avoid misgendering, the person I am trying to reach. Selfishly, I want to be saved from embarrassment. But also, I want to save the person I'm addressing the embarrassment and awkwardness of being misgendered--because that also brings an entire additional dynamic to the conversation. They are now in the uncomfortable position of needing to decide whether or not they correct me, and regardless of their choice, an additional level of discomfort been added to the interaction.
If everyone used pronoun identification reflexively, it would smooth those interactions. Rote business interactions would continue to be just that: rote. If I have to call my bank or the doctor's office or whatever it happens to be, I want it to be as quick and easy as possible and limited to the scope of the reason I called. I don't need additional levels of discomfort and frustration built into those interactions--and neither does anyone else.
So let's make everyday interactions easier, smoother, and more comfortable for everyone--regardless of how they identify themselves. It's such a small and simple thing to make one tiny part of someone's day a little easier.