Now, is it me, or does this go against the very idea behind the UFP, where interspecies co-operation is the key?
What could possibly give you the impression that this is one of the "key" ideals of the Federation?
There are logical reasons that individuals from all the member species of the Federation couldn't serve aboard the same ship. One that is brought forward frequently is they are unlikely to all breath the exact same gases, at the same pressure, or live in the same temperature and gravity range.
Trek fiction sometimes depicts Mister Spock's living quarters with the gravity system turned up, and the heat too.
For all we know, Humans couldn't live in the atmosphere on the vast majority of Starfleet's ships. 95% of all Starfleet vessels could have fully integrated crews, with a 140 species all serving together. And 5% of Starfleet vessels are reserved for the 8 or so species who breath a "Earth average" atmosphere.
Humans are among the odd ones out.
It's also possible that various Federation species could not serve together for psychological reasons. Or sociological ones. Medusians (not sure if they're a Federation species) can't serve with Humans, and Vulcans can only be in their presence wearing a special visor. Others might have no problem working with them.
Some of the Federation's specie wouldn't be able to serve aboard the Enterprise because they physically wouldn't fit in the corridors and turbolifts. Imagine an sapient species with the adults being the size of a elephant.
That was going to be my answer as well. Breathing the same gases, at the same temperature, in the same environment would be incredibly tricky. Alter the oxygen and nitrogen levels a few points either way and humans can't breathe.
Come to think of it, it's quite amazing to get more than 2 species on the same ship given the environmental requirements, even if a self-contained breathing unit was used.