No, they don't. I personally know a cis person who has their pronouns on hexbear set as they/them or comrade/them. Those aren't the pronouns they use in real life.
If someone assumes you're trans simply because your pronouns are set to something other than she/her or he/him, that's kind of on them. People are allowed to lie on the internet. Choose any pronouns you like, it doesn't matter.
I personally assume absolutely nothing about the cisness or transness of someone with "any" or "none/use name" as their pronouns of choice.
That mod was a piece of work, hilarious little saga all around
I understand your point of view more than most of the arguments I've seen against mandatory pronouns. So please take my comment as friendly, I'll do my best not to be a rude asshole.
How would you feel about (any) for your pronoun choice? That's functionally the same as not listing them, people can still choose which ones they want to use for you, but it still shows you're supportive of people prominently displaying their pronouns. That or you could consider maybe a neopronoun. I personally really like e/em/eir. They're nice and genderless, easy to use, and, bonus, a mathematician came up with them in like the '70s (I could have the year wrong and I refuse to look it up), not because he was trying to be trans inclusive, but because he hated that math books assumed their readers were all men and he wanted to include women in his writing. (Singular they was considered ungrammatical at that point.)