"Exist" is such an interesting choice of word. Clever in some ways, infuriating in others.
I guess, to the random layperson, it's pitifully emotive. But it actually lays bare the extremist character of the movement. I mean, the more predictable, honest choice would be "just want to live [our lives]" - and I imagine that's what many of the most vulnerable trans-identifying people who aren't aggressively activist would prefer.
"Exist", though, disregards any simple, shared needs with which everyone can empathise (to live with accessible facilities / freedom from prejudice etc.) to instead make "being validated as female/male by the random passerby" the thing that's absolutely fundamental.
Also (and I may be overreaching myself here, but I think there's some truth to this...) whereas "to live our lives" feels like it has a more positive focus - a meaningfully active, hopeful verb phrase - "exist" is all negative connotations of death, denial, negation etc., and passive as anything.
It also puts all the responsibility onto the interior perception of the person seeing the billboard: if you don't validate trans people, they may as well not be alive. It's gaslighty, and even irresponsible.
And so the more meaningful, essential and accessible needs lower on Maslow's hierarchy - needs that they'd argue themselves are still not yet met - just get swamped in this total overreach.
It's maddeningly self-defeating.