Okay, let me try and break this down...
Centaurs:
The thing about Centaurs is that there's at least two different "archetypes" to them that I can recall; the brutal, war-like and savage centaur, and the swift-running and agile centaur. I think also that centaur lore traditionally portrays the herd's shamans as literally born different to the rest of the herd?
With this in mind, what I'm currently leaning towards is a sub-race approach; one "warlike bruisers" sub-race, one swift-runners sub-race, and, most tentatively of all, a shamanic sub-race. Figuring out distinguishing traits for the sub-races is proving difficult, though.
Also, traditionally, centaurs haven't been fae in D&D so much as they've been monstrous humanoids who get on well with elves. For this reason, what I'm thinking of doing is giving them the option to add Fey Ancestry in a sidebar; this supports DMs who want them to be fae, as they were in 4e, or purists who prefer them as purely mundane monstrous humanoids.
Satyrs:
Hmm... I'm still struggling to find a niche for Satyrs, but, with the lore on them compared to fauns, your suggestion of more "primal/warlike" satyrs vs. more "tame" satyrs makes sense. The mythology amateur in me is now wondering if it's possible to add in the Ipotanes (horse-satyrs) in some fashion...
Hmm, I may have some ideas. I'll see if I can post them up here at some point.
Nymphs/Nature Spirits:
The big problem with nymphs in general is how to make them interesting. Generally, your nymph is just an uber-beautiful female elf... now, don't get me wrong, I love me some cute monstergirls and would love to explore better meshing that trope with D&D, but this isn't the kind of forum where it's safe to do so. THough, I think I recall seeing a homebrew about 5e racial writeups for the races from Daily Life with Monstergirl... moving right along!
The problem with doing nature spirits is pretty close to the problem with doing nymphs: how do you do them well? I mean, who honestly remembers Spirit Folk, which were literally "nature spirits in humanoid form" from 3e's Unapproachable East and Oriental Adventures splatbooks?
Heck, I could easily plop down elemental nymphs in my games by just reskinning & renaming the genasi as appropriate and giving them Fey Ancestry.
I'm not averse to it, but we need to come up with a way to make them actually interesting.
Dryads:
Here's the two big issues with Dryads, leaving aside balance; how do we make them more interesting than just "green-skinned she-elf druid/enchanter multiclassed", and how do we account for the whole "stuck with their tree" thing?
That's partly why I started with just the Hamadryad, which is a more powerful version of the Dryad from AD&D that was characterized by its ability to leave its tree:
http://www.lomion.de/cmm/hamadrya.php
Honestly, I'm quite a fan of 4e and Warhammer's idea that Dryads are fae tree spirits who, thusly, can shapeshift between more human and more plant-like forms, which would certainly be easier (and, in my opinion, more interesting) than trying to balance them having access to high-level druidic spells and abilities.
There was a Dryad/Hamadryad in 4e's Heroes of the Feywild that might be interesting to look at for a design basis, perhaps?
Fae Goblins:
Honestly, I agree; heck, probably the simplest way to get a "fae goblin" is to take the gobbo from Volo's Guide, slap on Fey Ancestry, and bam, you're done.
Tinies:
I really need to dig up the Heroes of the Feywild and the 5e MM before I can properly comment on doing these. I suppose the big thing to answer is "would Sprites and Pixies be separate races, or sub-races of the same Fairy race"?
Flyers:
For the Aaracokra, personally, I agree that it's underpowered for what it's supposed to do.
That said, I also can agree that flight should probably suffer some restrictions based on being heavily burdened or paralyzed because, really, how do you expect someone to fly in non-magical platemail?