Why are there pixies, etc. on the First but not the Source?

[Enkidoh]

On the other hand the First doesn't have Sylphs, Namazu, Ananta, Kojin, Vath, Vanu Vanu and even Moogles.

Moogles do exist on the First as well - they have blue poms and are seen even running a version of the Moogle Delivery Service (presumedly G'raha established it there based on the Source's one, but it still leads to questions as to how that can connect with the Moogle Delivery Service on the Source - presumedly it's similar to how retainer services are handled by Feo Ul, but that's a long shot).

Ananta and their lamia cousins not existing on the First makes sense though, given those were synthetic creations of the Allagan Empire same with the giants/gigas and so never had a counterpart form on the First when the Sundering happened (although it's possible some might have hitched a ride to the First via the Crystal Tower like the gremlins did). But as you said, a lot of beast races on the Source not being seen to have counterparts on the First can be just chalked up to being rendered extinct by the Flood of Light.

[Lumivyory]

The first had namazu like creatures in ronka or came to ronka via the Qatari or rat race before the flood. You learn that in their beast tribe quests

Moogles do exist on the First as well - they have blue poms and are seen even running a version of the Moogle Delivery Service (presumedly G'raha established it there based on the Source's one, but it still leads to questions as to how that can connect with the Moogle Delivery Service on the Source - presumedly it's similar to how retainer services are handled by Feo Ul, but that's a long shot).

Ah, totally forgot about these lol

And I suppose we could still encounter Source's versions of Pixies or Nu Mou when we venture into the new expansion, or it could be completely new beast tribes. For now we can only assume that they either don't exist or aren't in any area we have explored.
To be honest I'd like them to stay exclusive to the First, it's nice to have some differences.