I'm less 'Dominant Member Species' and more 'politics is hard, much harder when people are very different'.
Like, I'll parody the classic racist ranting.
Lots of Apiata are moving to Vulcan (or Space Goblins to Space Elf land if you prefer). Let's say they reproduce twenty times as fast as Vulcans. We'll stipulate that the Vulcan variant of logic isn't super persuasive to them, and they prefer to get their opinions from their queens.
Well, the obvious answer to this case is that the Vulcans and Apiata have significantly different environmental preferences- a Vulcan's idea of a nice environment is probably a rather disagreeable desert to the Apiata, so the Apiata have no desire to settle the favored planets of the Vulcans. And frankly, the Vulcan homeworld itself is kind of ass, environmentally speaking, being kind of burned out from old nuclear wars.
But that's specific; let's zoom out a bit and look at the big picture.
In addition, the Federation exists in part to delineate territorial spheres of influence that avoid situations where one member species ambitiously expands into space and blocks off another member species from having "a room of its own," so to speak.
So Vulcan and some of the systems around Vulcan are property of the Vulcans, effectively, and Vulcans have dibs on colonizing them. The same applies to the Apiata. The Apiata may have an advantage when it comes to expanding into previously unclaimed regions of interstellar space that are effectively
astra nullius (e.g. the Gabriel Expanse), but this doesn't prevent the Vulcans from having a degree of regional autonomy and being able to live in a society ordered according to principles that are pleasant to Vulcans.
Basically, to get the kind of scenario you're worried about, you'd need Federation Member A to start indulging in outright settler colonialism on the homeworld and any major colonies belonging to Federation Member B. That's not something the Federation's internal constitution is going to allow. And any species with a manifest-destiny attitude that it has the
right to claim all the real estate in space and leave other Federation members nothing other than the ground beneath their feet isn't going to be allowed into the Federation until it is disabused of this notion.
I don't think that those are crazy assumptions, certainly not beyond the stuff that the show has trotted out. Like, the Betazeds are telepathic, right?
So, like, now what? Do Vulcans just accept that from now on their gov will be made of Apiata (who outcompete Vulcans for Apiata votes) and pursue policies that attract Apiata votes, even if those are illogical by Vulcan standards? Do they go alt right and start chanting mathematical formula that round to 'they will not replace us'? Do they just retreat into their holodecks. or maybe do a Science War Crime?
Or maybe Star Fleet doesn't allow the migrations, Star Fleet ICE? Kick the can down the road a bit? Do a three fifths compromise deal, where only so many Apiata votes equal a Vulcan vote? None of those sound remotely in character, but...
I'm not postulating a deliberate and sinister Apiata Quiverfull here, just species being different and that fact giving one or the other an advantage using basic "one sophont, one vote" rules, and the control of Star Fleet/The Federation standing in as a limited resource in a mostly post scarcity world.
Yes, but you're also making the tacit assumption that swarming outbreeding is a normal behavior for certain sapient species and that they will naturally populate all of Federation space with swarms of themselves?
One could equally well argue that for natural sapient species that have independently figured out warp drive, uncontrolled impulsive breeding that the species collectively refuses to keep within limits so that it can remain within an agreed-upon territory for the foreseeable future is a huge
disadvantage. Explosive-breeding swarm species, if they develop sapience at all, are liable to overrun and exhaust their homeworld too quickly to discover warp drive.
Arguably, this is one of the reasons the Federation
doesn't uplift prewarp species. They don't want to find themselves in a position like the Citadel from Mass Effect did with the krogans. The Citadel, having actively encouraged the krogans to reproduce rapidly and take an active role in interstellar affairs, found that the krogans became a destabilizing threat- and one with an entitlement complex. Leaving the krogans alone until their culture stabilized somehow would probably have been beneficial to the Citadel, if not for the urgent short-term need for janissaries to fight the rachni.
Like, presumably Data votes after Measure Of A man, and dude can build as many of himself as he feels like and survive/thrive anywhere, oh and also he lives forever. He's a cool guy, but are he and his buds so cool that we want them running the entire Federation? If not, at what point do they stop voting? Do they have an obligation to listen if someone tells them that they aren't allowed to have any more kids? Who would possibly make that call, and by what authority, but if no one does how do you stop the Data Fork Block from being the majority of all sentient life whenever they feel like it?
If this starts to look like it's actually going to become a problem, presumably the Data Fork Block can (under TBG-style Federation rules) be treated as a 'Federation member' or similar entity in its own right, and negotiate rights to a certain share of Federation territory and resources, without being extended unlimited rights to copy themselves indefinitely and consume everything.
I get that the above sounds a lot like human racist talk, which, fair, but I do think Star Trek's variations are way more extreme than anything the real world allows. I can think of lots of answers to the examples above, but they all feel like much more politics than the Federation is depicted as having, and like they'd be the start of arguments, not the end of them. I believe that there would be lots and lots of civil wars, mostly caused by populists stirring up resentment of other races who they see as unfairly advantaged by a system that can't possibly deal with this many incredibly different races.
The Federation basically just handles this by negotiating agreements that require all the Federation species to respect each others' boundaries and refrain from activities that might threaten or destabilize those boundaries in an unwelcome way. If the Vulcans feel like inviting a billion Apiata to live on Vulcan, and a billion Apiata want to go, that's fine... but having a billion Apiata move in on the Vulcans when there is no obvious pressing need for them to do so is undesirable. If some disaster overtook the Apiata homeworld and they had billions of refugees to be somehow transported across space, that might be different, but then that would be a major logistical challenge in its own right that would be dealt with separately.