"Sometimes I think the "gender neutral" or "non-binary" types are actually more hung up on narrow stereotypes than the general populace."
That comes from outside pressure. If you're not a binary cis person, some people won't believe you about the existence of your own gender unless you fit a very narrow stereotype that they've come up with. So many nonbinary people are afraid that if they experiment at all with anything considered "too girly" or "too boyish", people will misgender them.
Even binary trans people have the same problem-- there are trans men who are afraid to look "too girly" because people won't believe that they're boys, and trans women who wouldn't feel comfortable looking like me (a cis woman) because people wouldn't believe they were girls. It's not because they're hung-up on narrow stereotypes, it's because other people push them into the narrow stereotypes and make it difficult to go outside them.
Sometimes this is relevant to names (if you're nonbinary and your name is too "masculine" or "feminine", people will constantly have to be corrected about your pronouns so it might be easier to change your name to something that's already considered "neutral" even if you actually prefer a name that isn't stereotypically neutral), but not usually relevant to naming babies since you don't know the baby's gender. Most people name the baby assuming it will be cis, with the assumption that they'll change their name later if it doesn't fit.