Here's an out of the box answer: how about starting out with no playable aliens? No need to go gonzo right out of the gate. Some of the best space science fiction sagas--Dune, Firefly, BattleStar, The Expanse, to name a few--were successfully told without any alien civilizations at all.
You could start Starfinder Society season one with several human factions in an exploration race. Humans might already have different colonial planet origins or ethnicities and maybe even levels of cybernetic enhancement and gen-engineering. Later seasons could add some encountered aliens and occasionally lift "boon restrictions" upon their playability. I would like to see alien species, when available, as having relatively small and double-edged bonuses and penalties that revolve more around situational terrain bonuses and damage vulnerabilities than stereotypes or inherent class superiority so as to not pigeonhole aliens to self building class "choice".
Admittedly, the above idea may be unpopular as other commentators already have their own alien ideas that they like. Alternatively, you could set core races not as alien races, but classifications of life. Instead of Humans, there is a "Humanoids" category and the player chooses abilities from this list and can name the character's species at the end. In place of Dwarves, there is another list for Lithivores and Silicon based lifeforms; further lists for mechanoid lifeforms, and so on. A number of turn based Space Strategy games like Galactic Civilizations and Master of Orion allowed players to design their own alien in such a way.