The Federation we're playing as is still to small to represent the federation that was shown to exist in TNG. We're not that many years away from it either if anything we're not affiliating and creating new members quick enough.
The problem here, as I and others have pointed out, is that the quest had us start out with just the original four members in 2300.
Canonically, the Federation had already been gaining members since at least Kirk's time (we saw the ratification of the Coridons in the episode "Journey to Babel"), and probably much earlier (the Rigellians and Caitians were probably intended to be members rather than just affiliates when we were introduced to them).
What probably happened in the canon timeline is something like:
1) The Original Four found the Federation in 2161. The Rigellians, Denobulans, and a few others immediately affiliate.
2) A decade or two later, say in 2180, Rigel joins as a full member.
3) A decade after that, Ferasa or Denobula or whoever does the same.
4) By Kirk's time in the mid 2200's, the Federation has built up a respectable roster of about 8-10 species. The era of rapid expansion in the mid to late 2200's causes that number to start growing more rapidly, with new species like the Coridons and the Deltans/Betazoids being added every 4-5 years.
5) By Picard's time a century later, the Federation has so many species in it that the council can do a reasonable impersonation of Star Wars' galactic senate. The Federation is so big that the ratification of new members is considered mundane, someone gets ratified almost every year and it barely has any impact on most Federation citizens.
Basically, this is the snowball effect in action. When your federation only has four members, adding a fifth is going to cause a massive restructuring. You can only add new members once every decade or so, or else the political chaos will tear the union apart. Once you've got ten members, however, adding an eleventh isn't nearly as difficult, and you can do it faster while rocking the boat less. Once you've got a hundred, its a drop in the bucket.
What bothers me about this quest isn't that we're adding members too fast or too slow
right now. Its that we transitioned too quickly from "four members, considering maybe possibly adding the Amarki as a fifth" to "twelve and counting." The five year moratorium paid lip service to acknowledging this problem, but our timeline is still pretty insane compared to the probable canon one.
Its something I'm okay with handwaving with some thin justifications, the same way we kind of handwaved Starfleet being so ridiculously tiny at the quest's start with some thin justifications. To Boldly Go follows a tried and true game design paradigm in which the player(s) start out in control of a very small faction, and reap the fruits of exponential growth in response to their successful projects. The rate of that exponential growth is very well paced to make an enjoyable and rewarding game experience. However, it is not very well paced to realistically portray the expansion of a space federation.