Well, it's not really about us. It's about USians, and in the US the community brand that won out is 'Latino'. But yeah, if you want to make that gender neutral I can't see what's wrong with Latin People or Folk, as opposed to Latinx Folk or People.
Latin American as a term was tied to the language of academia and diplomacy. It started as a French geopolitical delusion, continued onwards as a sticking point in hispanic literature, and only came to really include Brazil in the early 1900s due to American geopolitical reality. Since then it described foreign peoples, not a community within the United States. I'd wager that's why it didn't catch: a black person in the US can fill a form referring to themselves as African American - but the mexican person won't find any forms with Latin American as a matter of historical circumstance.