But to clarify your concern (ignoring the butt concern, for which I am unqualified and ill-advised to offer counsel), you are objecting to this sloppy construction: “A man was seen leaving their nose prints on the window of Lamar’s Donuts shortly before a dozen donuts went missing.” Instead of “leaving his nose prints,” right?

I have to say, I’ve seen it occasionally, but not frequently. I’ll look for more instances, and if I see enough, I’ll write a post. In the meantime, folks, make your pronouns match their antecedents (the nouns to which they refer), e.g., man=his, woman=her.