Okay, "quick" review (warning, this is going to be a big post, as this is a big UA, and I'm going to go down everything individually).
Let's start with the 4 "new" races, the three dragonborn races (which should just be subraces of a larger base race, IMO) and the Draco-Kobolds. I'll be grouping together the three Dragonborn, as they're very similar with very few differences.
These Dragonborn are pretty much the same as PHB Dragonborn with the race/lineage changes from the Fey Folk UA, better breath weapons, and a powerful race/subrace specific ability. The breath weapon changes are very welcome, especially being capable of being used as part of the Attack action (extremely useful for melee Monks, Fighters, Barbarians, and Paladins). The only thing I'm not on board with here is the race-dependent breath weapon shape, which I feel should still be tied to the Draconic Ancestry. Chromatic Warding is bonkers useful, but I don't know if it's too powerful. I would probably limit it to one minute, and possibly make it concentration. Immunity to a very common damage type at 3rd level (even if it is temporary) seems too potent, IMO. Metallic Breath Weapon is very useful, and I love that you can use it for different ways when you need to. Both are useful in a variety of circumstances, and I don't think either are too potent. Gem Flight seems fine, if not a bit strange to have a dragon wing made of gems. So, yeah, these changes are very welcome.
The Kobold is . . . strange. I'm not against redoing Kobolds, as they're quite strange and unbalanced in terms of mechanics in the Volo's version, but I don't think this is the way to do it. I like having a variety of draconic traits that you can choose from, but this just seems like a strange way to do it. I'd give the free Sorcerer Cantrip to the base race, and then move the Roar to an option from Draconic Legacy. The tail seems like a strange choice of natural weapon, I would have guessed claws or even a bite before a tail-whip. Anyway, something about this isn't working for me. I'm not sure what it is, though.
Next, the Feats. I don't have any comments on Gift of the Chromatic Dragon, it seems very potent and fun, other than possibly allowing it to work with natural weapons/unarmed strikes. Gift of the Metallic Dragon is . . . strange. I know Metallic Dragons are supposed to be good, but not "cure wounds", protect others, goody-two-shoes good. I'm not sure how I would change this . . . but I would change it. Maybe some minor shape-shifting ability? Metallic Dragons do have a Change Shape ability, so something like innate
Alter Self could work. Gift of the Gem Dragon seems too psionic and not draconic enough. Gem Dragons are psionic, but not
just psionic. They're also dragons, and nothing in the feat (other than the name) screams
dragon to me.
Finally, the Spells. There are quite a lot of these, and I like that there's so many of them. I've been saying for quite awhile that we need more dragon-based spells. Draconic Bloodline Sorcerers are currently just elemental blasters with a touch of dragon theme, with barely any dragon-spells to choose from. This changes that a bit. I'm just going to touch on each spell a bit to keep this short and sweet.
Draconic Transformation seems too weak, especially when compared to Tasha's Otherworldly Guise. Both grant 40 feet flying speeds, are concentration (up to 1 minute), require 500 gp worth of material components that aren't consumed, and take a bonus action to cast. However, Draconic Transformation takes a 7th-level spell slot, instead of a 6th, doesn't grant any immunities or even resistances (while TOG grants 2), no condition immunities, no AC boost, no special magical weapon boost, or Extra Attack feature. In return for not having all of that stuff and requiring a higher level spell slot, this spell only gives the caster a mere 30 feet of Blindsight and a bit more powerful
Dragon's Breath ability. This is just not enough for a concentration 7th-level spell. At the very least this spell should also let you add your Spellcasting Ability to your Armor Class in order to bump up its power and usefulness more, and maybe even give your resistance/immunity to some draconic damage types. I love the idea of this spell, it just needs more oomph in order to be worth taking.
Fizban's Platinum Shield, on the other hand, is probably too powerful. It gives you (and/or an ally) resistance to all the main draconic damage types, half-cover, and Evasion. That's really potent, even for a 6th level concentration spell. If you want to fight dragons/mages that force Dex Saves a lot, this is the spell to use. You will have a +2 to the saving throw, and take no damage on a success and half on a fail, and probably also have resistance to the damage that you do take. (This spell also perfectly demonstrates why I have banned the Order of the Scribes from my table. Being able to swap out those damage resistances for practically any others that you want is just way too crazy useful.) I'm not sure how I would change this, but it probably would involve making the spell be a higher level, or possibly add a reaction to either the resistances or Evasion feature.
Flame Stride is fun. Really fun. No real comments, other than maybe, possibly raising the damage a bit. 1d6 seems really low for a 3rd level spell, even if it affects anyone within 5 feet of you. Maybe change this to 2d6, or 1d10. This doesn't seem very draconic, but I don't really care. It's a cool, fun spell. Icingdeath's Frost is also fun. It's basically a more powerful
Frost Fingers with a really potent debuff feature. We need more cold damaging spells in 5e, so this is also a welcome addition to the game.
Screw Fairie Dragons. Seriously. Screw Fairie Dragons. I hate those bastards. My Hexblade Warlock has an undying hatred for them, due to a particularly annoying encounter involving two of these rainbow cats with wings that almost annoyed him to death. Don't mistake their puny 1 damage bite attack for them being weak enemies. Trust me, if you throw a couple of these abominations at a low level party that can't see invisibility, you're going to give them the eternal hatred for them that I have for these glorified pseudodragons. Nathair's Mischief can go to the Nine Hells of Baator, for all I care. (In case you like fairie dragons, don't take offense. It's not your fault that they charmed you with their draco-feline behavior. Please seek help and recover as quickly as you can.)
Raulothim's Psychic Lance is really, really good. I would be concerned about how powerful it is, except for the fact that it is most potent when you know their name. I really like this spell, even if it is more psionic than draconic.
Summon Draconic Spirit is good. I don't have any real comments here. I would like a higher level spell that lets you summon a real dragon, though. It would probably have to be 8th or 9th level, but it would really be awesome to be a Draconic Sorcerer that summons a real adult/ancient dragon to aid them in combat instead of just a spirit of one. Otherwise, this looks good.
Overall, I like this UA. I really hope that this comes out in a Draconomicon, which seems likely at this point, as we really need more Dragon options in Dungeons &
Dragons. I would love to see more subclasses, like a Draconic Warlock Patron, Dragonrage Barbarian, and Dragon Druid, but this is a step in the right direction. I also hope they make more spells than just these few, as Tasha's disappointed me with having less than 2 dozen spells in it (including the reprints).