But don't languages with neuter pronouns usually reserve these pronouns for a few objects (or all objects, in English's case) only and no people?

Pronouns, yeah.

English does not have grammatical gender, and "they" was already neutral in the past, and entered colloquial usage for hypothetical people, so it is a whole different package.

Dutch and Scandinavians have merged or are in the process of merging male and female genders (a phonological evolution, not an intentional one, just to make it clear), leaving a common gender along with neuter, so pronouns are among the few places of contention.

German is a whole thing, as the formal way to refer to someone and the plural don't have gender, so binary people don't really need gender-neutral language. So the small minority of people of non-binary people go with different approaches. Some use the neuter, some don't.