Why is "folks" commonly used as a gender-neutral term for "people" when "people" is already gender-neutral?

First of all, this is absolutely a usage that is particularly notable within LGBT and social justice communities, and there is definitely some sort of gender connotation. People who do not spend a lot of time in these communities may not be familiar with it. While "folks" may be on the rise in general, it is disproportionately so in these specific contexts. The "folks" phenomenon comes up often in internet parody.

My personal speculation is that this usage originated with demographically descriptive adjectives ("trans folk/s", "black folk/s") for a couple of reasons. Some phrases, even though absolutely neutral, with demographical references in them, have become latently politically charged in the eyes of out-groups even though there is no specific connotation (such as "black people"). I think the use of "folk/s" is to avoid that phenomenon. In fact, this also allows the speaker to avoid saying something like "these people" referring to a set of specific people, accidentally sounding like they are referring to the entire class the person is in (even though "folks" isn't any more specific than "people" in this regard-- "persons" is rarely used in speech-- there is not going to be the hair-trigger association that something like "these people" has). Also consider use in a greeting: "Hey people!" sounds blunt and almost aggressive to some. I'm not able to explain why, but I know that it can and does offend certain people. Almost like the speaker is a teacher yelling for people's attention. "Hey folks!" avoids this.

The OP states confidently in their question

Why is “folks” commonly used as a gender-neutral term for “people” when “people” is already gender-neutral?

The term folks, originally an Old English term used on both sides of the Atlantic, has not seen a recent surge in popularity because it is a better alternative to the gender neutral people, but because it sounds friendlier, and more welcoming than "people".

Two Google Ngram below showing the trends for folks in the American English corpus from 1900 to 2008

and in the British English corpus from 1900 to 2008

The former Oxford Dictionaries Online, Lexico, does not mention anything about folks being a friendlier gender-neutral term compared to people.

Lexico
1.1 folks

Used as a friendly form of address to a group of people.

• ‘meanwhile folks, why not relax and enjoy the atmosphere?’
• ‘I think we've located another point in our musical journey here, folks.’
• ‘Mark this day on your calendar, folks, because it is a rare occasion indeed.’