Lots of species we've seen have a gimmick - they live longer or have exceptional biology in some way - and I think part of that, along with some truly excellent timing, played a role in Human dominance of the Federation, and also by comparison, why the Klingon Empire got so big as well.
It's about sex/reproduction (yes, really) and it's about timing of United Earth's rise to prominence in local space. Hear me out.
Let's go back to the time of Enterprise NX-01 in 2155 and consider local space at large. Humanity was largely Earth and Earth-orbit based, but had substantial infrastructure on the Moon, a few thousand colonists on Mars (that was in the middle of terra forming) and a base at Jupiter. A couple dozen supply ships ran long duration / slow speed cargo runs. There was presumably a settlement at Alpha Centauri but humanity was mostly bottled up inside Earth-Moon orbit. This is true of almost other species out there at the time, with a few notable exceptions.
The Vulcans and the Andorians were the two most prominent and advanced local space powers, with the Tellarites a somewhat distant third. Each controlled the entirety of their home system and several other nearby systems. But both kept their fleets close to home because the Galaxy was quite dangerous at that time. The Vulcans had the best technology and best ships. The Andorians were a little behind them but seem to have more ships. But neither species did much besides guard their territory. The Tellarites were presumably on this level too, with ships that were a little slower and a little less advanced, but still well ahead of United Earth.
This exists in contrast with the Denobulans, who were a bit further away from what the "core" of the Federation would be in a few years. They had advanced space technology for many years, a martial history to them (a series of interstellar wars) and a huge population. But they only lightly cooperated with interstellar politics. They sent representatives to various functions and made their presence known, but despite their technology, which was far more advanced than United Earth, also stayed within their own star system, very much like United Earth and unlike the Vulcans or Andorians. The Denobulans represent what would be fairly typical for future (but not founding) Federation worlds - despite their participation, their home system is where they stuck to.
So what was the advantage of humans? I think it is timing, political necessity combined with the nature of human reproduction (seriously). The Vulcan race is lives on average around 200 years. They go into Pon-farr every 7 years. The Vulcan population seems to not grow all that quickly. They don't suffer a lot of natural deaths, but also don't naturally multiply. They seem to exist at exactly population replacement. And we know from Star Trek (2009) that destroying Kelvinverse Vulcan left about 10,000 survivors of the species, which meant that at least in that time line, 20 years after the divergence, one planet supported almost the entire race. Then there's the Andorians who have very complicated mating rituals involving 4 sexes. That would mean population replacement would require twice as many "inputs" to generate the same "output". It would be very difficult to sustain such a population, much less grow it, especially if Andorians lived as long as human. Some of the books have dealt with this. There just aren't that many Andorians. They live on a moon of an Ice Giant even they describe as harsh and they evolved there.
Humans, by contrast, live for about 120 years by the 24th century, and probably around 90 on Average in 2155. Compared to Vulcans, the window of time in which reproduction can occur is very, very wide. Compared to Andorians, it's simpler. And we've seen the results of it: a lot of two and three children human families in Star Trek. A lot of children in general, whereas it seems many Vulcan families have one child. Compared to Vulcans, humans are also more sexually gregarious... in many ways closer to Denobulans than anyone else. As we see in the 24th century, they can interbreed fairly easily (with medical help sometimes) with almost any other humanoid species and that those species don't or can't interbreed with each other.
What I think it comes down to is this: in 22nd and 23rd centuries (likely continuing into the 24th), the human population exploded. A population of 7 billion in 2155 (fewer than we have today, accounting for projected decline in birth rates as predicted by the UN) growing at a rate of 1.1% (the rate of growth in 2021), would grow to 21 billion by 2255. By 2355, it's 63 billion. If you increase the rate of growth (assuming assist by technology, needs of colonization, social changes) to 2.1%, there are 55 billion humans in 2255 and 446 billion humans by 2355.
I think the human population grew to be absolutely colossal simply because, well, humans reproduce far easier and more often than Vulcans and Andorians. So why didn't other species gain such prominence in the Federation? Humans were first. United Earth was technologically and politically behind the Vulcans and Andorians in 2155, but well ahead of many future Federation members who would join over the next 200 years. Other species who joined in the Federation in the latter part of the 22nd, 23rd and 24th century would be playing catch up to an institution that had a massive number of human beings operating at every level of it from the start. And they simply may have had smaller populations to begin with. According to Memory Alpha, the Trill joined the Federation in 2285 with a population of just 650 million. That would be about 1/28th of the human population at that time. They just didn't get into interstellar affairs early enough with a big enough population. Another word would be "momentum".
Another species had that kind of "momentum" by the way: the Klingons. Like humans they reproduce easily. Like Vulcans they live a very long time (though few make it to that age). And like the Vulcans, Andorians, and Humans from a certain perspective, they were active in interstellar affairs early. And they built what was, in 2155, the largest known local Empire with technology more advanced than the Vulcans in some ways. And by 2375, they had the second largest empire in local space. They also fielded huge, single species military forces and took enormous causalities. I think the Klingons successful for the same reason: they got "on the scene" early, and they had big families with lots of kids, who in turn had big families with lots of kids. It wouldn't surprise me if by the late 24th century there were hundreds of billions of Klingons in the empire either.
You combine the above human population growth with the fact that the Andorians, Tellarites and Vulcans only mutually trusted Humanity, and you got the makings of essentially perpetual human dominance of the Federation because later joiners - most of whom will be like Denobula and keep close to home in any event - will never catch up in sheer numbers, if they have anything approach it to begin with.
