I believe it is possible to focus on humanity while not being human or earth-centric. It's all in how you tell the story.
All of Trek has always contradicted it's multicultural philosophy with the way stories are told and non-humans are presented. It's subtle things such as describing aliens. TOS was especially guilty of this. Spock, a Vulcan, never called something human as being "alien in origin." Alien was always used to describe something non human or not from Earth. Everything was an Earth colony.
You can create a story, such as Star Trek IV where Earth (and therefore humanity) is threatened yet the dialog doesn't have the characters singling out Humans specifically. It's the 24th century (well, 23rd in TOS era) and humans are spread all over the quadrant and there are millions of non-humans living on Earth.
Sure, maybe Q had something of a special interest in humans. A botanist may study a wide variety of plants and still can have a favorite flower. Q might just have some special fondness with humans which is why he singles them out.
Regardless, there are still ways to tell the story that gets your human audience involved and engaged while creatively treating the galaxy as a cosmopolitan place instead of always being human first.