Why?
Why not? Evoker is fun and has mechanics unavailable in other classes, and provides a class fantasy that other races would be able to mesh with. Dracthyr are just another race among many, and it feels silly that they wouldn’t be able to adopt some of the skills other races are capable of.
I am okay with the idea of all races being able to play every class, but considering Blizzard likely isn’t going to go in that direction any time soon (making Druid and Demon Hunter forms for every race is a scary concept, and the lore implications of a Forsaken Paladin currently seem too great to just overturn without a thought), we’ll just have to do what we can to fit it into the current template of things.
How would it work?
It’s all a matter of creativity.
Because the Dracthyr and the Evoker class are intertwined in several ways, it would require some reworking some of the core concepts to make it usable for other races/classes.
Glide, Tail Swipe, and Visage would become Dracthyr’s Racials. Their Tempered Scales could also be, though a nerf would be in order to prevent it from becoming OP when playing a Tank.
‘Dragon breath’ abilities like Fire Breath could be handwaved one of two ways; Non-Dracthyr characters can either just straight up breathe fire (a-la Brewmaster’s Breath of Fire ability), or conjure an ephemeral dragon’s head to breathe it for them (a-la Mage’s Dragon’s Breath).
Abilities that imply/require wings such as Hover or Deep Breath can have the character conjure magical wings behind them. The color could be customized similar to unique Demon Hunter customizations. Since I’m not Trust Level 3, just look up ‘FFXIV Summoner Limit Break’ to see what I mean. The wings aren’t part of the character, they’re an extention of their power.
It will be easy enough for Dracthyr to occupy other classes. Maybe even Demon Hunters would be open to them since they already have wings? 
Lore Explanation?
Considering that the story currently doesn’t point in one direction or another regarding this topic, we’ll have to make up our own hypothetical future justification.
Here are two that jumped to mind immediately when thinking of the topic:
Hypothetical Future 1 (Likely the more plausible):
With the help of the depowered Aspects, we defeat Iridikron and the other Incarnates, finally bookending the Primalist threat once and for all. For whatever reason, Iridikron’s presence was preventing the re-empowerment of the Mother Oathstone, and with him gone, they are free to wield their power. However, during the events of Dragonflight, the Aspects (minus Nozdormu, probably) discover that they never actually needed their powers to help safeguard the future of their world, and decide to give the power to the mortal races, who have acted as defenders of Azeroth in their absence since the end of Cataclysm.
This manifests as many mortal races now being able to wield the power of the Evoker class, as the power of the aspects ‘spreads out evenly’ among them.
The Dracthyr on the other hand, having come to terms with the circumstances of their creation, learn how to proliferate their kind in the absence of Neltharion without tying them to the dragonflights. The resulting creations are dragonkin, but do not necessarily always have Aspectral powers, similar to the Drakonid that also occupy the Dragon Isles.
Hypothetical Future 2:
The Aspects are successful in stopping the Primalist threat, though at a terrible cost. By combining what remained of their power and life forces in a manner similar to the Dragon Soul, the champion destroyed Iridikron and the Incarnates, though as a result the existing Aspects ceased to be (except for Nozdormu).
The use of this new Dragon Soul suffused Azeroth with this Titan power, and Tyr, who somehow became alive again through his weird homunculus body, teaches the mortal races to wield this new magic as Evokers.
The Dracthyr no longer innately have their powers in the absence of the Aspects, and thus must learn to be Evokers just as any other race. They can choose to take any path in life they wish, however.
What races would be able to become Evokers?
Because the Aspects’ powers derived from Titan influence, the main races who would be able to wield the power of the Evoker would be descendant from Titan creations, with a few exceptions.
Humans? 
YES. The Humans of Azeroth are probably the most famous example of the development of races being influenced by the Titans, with much of their society being built around the stories of Tyr, who helped empower the original Aspects. Also, almost every Dragon’s visage form is Human, or very close to it. No-brainer.
Dwarves? YES. Though they aren’t as close to dragons as other races might be, the Dwarves have proven themselves to be quite important to the history of the world, especially when it surrounds Titans and Titan Watchers. They would easily be able to pick up the tenets of Evoker.
Night Elves? YES. With how close they are to the Emerald Dream and the Aspects (especially Y’sera), this is a no-brainer.
Gnomes? 
UNSURE. While the Gnomes certainly have Titan history, theirs was never a history connected to the more naturalistic parts of the world. My gut says no, but I feel like it would be inconsistent to deny it.
Draenei? 
NO. Being as otherworldly as they are, they have little to no connection to the Titans as we know them, including Dragonflights. Sorry, Draenei.
Worgen? UNSURE. Would their ‘wild side’ mesh well with draconic power? I feel like they may have made too much of a sacrifice to maintain their humanity for them to make any real use of Evoker, despite their Titan ancestry and newfound connection to the Dream. Might be a bit of a stretch, though.
Pandaren? NO. Their development on Azeroth was almost entirely separated from the dragons, and I feel like the place the Dragonflights might occupy in their lore is already filled by the August Celestials.
Orcs? YES. I was hesitant at first, but the existence of Thrall gives them a pass.
Undead? UNSURE. I couldn’t think of any real reasons for or against Undead being Evokers. Would the Dragonflights have a problem with the Undead wielding their power? Would the powers of undeath and Order clash at all?
Tauren? YES. Tauren are just about as close to nature as the Night Elves, though in a different way. Why not?
Trolls? NO. Unlike the Undead and the Zandalari, their distinct ties to the Loa of Death might make things complicated.
Blood Elves? YES. What is Aspectral Power but just more magic to feed off of?
Goblins? UNSURE. With Goblins, it would probably be handwaved in anyway, for the exact same reasons why they’re able to be Shamans despite being completely irreverent of the natural world.
Void Elves? NO. Void and Titans don’t really mix, and we do not need Infinite Evokers running around.
Lightforged Draenei? NO. For the same reasons why they can’t be Shamans, Mages, Warlocks, or Monks: The Light is all that they are.
Dark Iron Dwarves? YES. For the same reasons as Dwarves.
Kul Tirans? YES. For the same reasons as Humans.
Mechagnomes? NO. Gnomes, but the reason why I was pensive is turned up to 11. No way.
Nightborne? UNSURE. Arcane and Dragon magic are two sides of the same Order coin, though it might be awkward to handwave a connection to the Dragons where there didn’t exist one for 10k years.
Highmountain Tauren? YES. For the same reasons as Tauren.
Mag’har Orcs? YES. For different reasons than the Orcs that I will not be elaborating on.
Zandalari Trolls? YES. What are the Dragonflights but bigger, fancier Loa?
Vulpera? UNSURE. I literally don’t care whether this race lives or is wiped off the face of Azeroth tomorrow, and as such do not care to research their origins or why they would want Dragon magic.
What classes should be available to the Dracthyr?
The classes available to Dracthyr is mainly just what world events they were present for, which is to say, not many.
Warrior? YES. Every race can be Warrior.
Paladin? YES. For reasons similar to the Tauren. The Tauren get Paladin in a hand-wavey way, with their worship of the sun manifesting similarly to the Light. With the Dracthyr, it could be cool Titan magic, as much of Tyrhold is animated by a power similar in appearance to the Light.
Hunter? YES. Every race can be Hunter.
Rogue? YES. Why not? The Blacktalon are basically a group of rogues, and there are Dracthyr in their ranks.
Priest? UNSURE. Every race can be Priest, but I don’t think it would fit well with Dracthyr. Void clashing with Order, worship of Light being uncharacteristic.
Shaman? NO. After what the Primalists did? I’d be surprised if anyone on Azeroth practiced Shamanism after the events of DF.
Mage? YES. Blue dragon magic is basically just Arcane Mage anyways.
Warlock? NO. Again, Titan stuff and demons/Void don’t mix well.
Monk? YES. Most classes can be Monks, and their theming is similar to that of the Evoker anyway.
Druid? UNSURE. Don’t get me wrong, I would love Dracthyr Druids. Imagine the cool dragonkin shapeshifting forms. Their connection to Ysera and the Dream also makes it make sense. But for some reason, Blizzard hates giving races access to Druid, so I don’t see it happening.
Demon Hunter? NO. You know why.
Death Knight? YES. It makes sense, as every other race can be DK, but… where is Bolvar? Is he cool? Is he still raising new DKs?