Every race/faction should have a starting world/base with some exclusive tutorial content and then it's out into a sandbox galaxy with faction neutral missions.
Access to certain systems would then be restricted for certain species and ships unless they have advanced in diplomacy enough.
That's basically how it is now
And honestly, I don't really mind. That's pretty much how Trek has always worked. TOS was Feds vs Klinks vs Rommies vs Random Aliens. TNG added Klinks to the Fed side...except when they weren't. Also added a few new enemies: the Cardies, Ferengi, Borg, etc. DS9 was the same way. Voyager was everyone vs. Voyager. ENT was Earth + Vulcans vs All, until the end (and even sometimes Earth and Vulcan fought!). The only real difference between STO and TV Trek is that in TV Trek diplomacy was an option, and the preferred one at that.