Good day to you all. This is my first attempt at a homebrew subclass. My goal is to create something with a strong theme; slowly becoming more like a true dragon, but on a mechanical level I want to find a way to make breath weapons viable beyond level 2. The dragon theme is hardly original of course, but I would appreciate any comments or thoughts you may have. Is it too strong? Too Weak? Too unfocused?  Would it be simpler to forgo the race requirement and give the breath weapon as an ability at 3rd level? 

Barbarian

Path of the Dragon Within

Special Restriction: The path of the dragon may only be walked by the Dragonborn. You must be a Dragonborn to select this subclass. This includes the Dragonborn in the PHB and the Ravenite and Draconblood variants in EGtW

The clans of the Dragonborn are rightly proud of the strength, culture and heritage. Many celebrate their freedom from draconic tyrants dethroned in ages past, others take pride in their association with their mighty forebears. There are some however for whom this is not enough. Warriors who know, in the very core of their being, that the Dragonborn are but a shadow of what they could be. By force of personality and the majesty of their rage some warriors are able to tap into the power of their ancestry and take the first bold steps towards dragonhood.

Fire & Fury

The storm within your soul rages harder in you than in others of your kind allowing you greater use of your breath weapon. Starting when you choose this path at 3 level you have a number of uses of your breath weapon equal to your proficiency bonus. You regain all uses on a short or long rest. Additionally, while you are raging, you can use your breath weapon as a bonus action. Using your breath weapon in this way does consume one of your uses.

By Fang & Claw

Your body is becoming more like that of a true dragon. Beginning at 6 level your hands grow sharp claws. These count as weapons, with which you are proficient; however you may still use your hands as normal including the ability to wield other weapons or shields. Your claws have following properties.

Name               Damage         Properties

Dragon Claw:  1D6 Slashing  Light

 Additionally, you are able to make a bite attack as a bonus action. This attack is a melee weapon attack; it deals 1D6+Str Slashing damage. If you are not using any weapon other than your claws and you use your bonus action to make an offhand weapon attack with your claws you may make a bite attack as part of that same bonus action.

Monarch of the Sky

The wings of the dragon are yours by right. By the power of your draconic soul you may command the universe to acknowledge this fact, for a time at least. Beginning at 10 level you may manifest a pair of dragon wings. While active you have a flying speed of 60ft. Your wings last for 10 minutes or until dismissed by you (no action required) or until you are incapacitated. Once activated you may not use this ability again until you have completed a short or long rest.

Dragon Re-Born

You have walked far down the path of the dragon and are now almost more drake than mortal. Starting at 14 level, when you manifest your dragon wings they last for one hour, or until dismissed or until you become incapacitated. Additionally, your inner elemental power is growing. While your wings are manifest your breath weapon does an additional 2D6 damage.  

After doing a brief skim of this, I see a few problems with this subclass and sorry that I jumped around a bit while explaining opinion;

• Race restricted: You did bring this up that it might be a bad idea and I would agree. There is a reason that WotC dropped this idea after the sword coast adventure guide. If you want,  keep the flavor that this path was originally developed by dragonborn clans, but the player has learned their teachings.
• Monarch of the sky: Even though I do like the idea of a dragon themed barbarian gaining the ability to fly, this is too much for a 10th level feature. Even the totem barbarian doesn't get that until 14th level, and is limited by needing to rage to fly. Even the draconic bloodline sorcerer, can't even do this until 14th level as well. If it was me, I'd integrate this feature with your 14th level then make the 10 level be something like a draconic presence feature while they're raging.
• Fire and Fury: I do think that you should change the wording of this feature so a non dragonborn can use a breath attack, but then a dragon born can have their's enhanced by this feature. This isn't my main issue with this tier, my issue is that there is no feature that changes how is used, this is a staple of barbarian subclasses in my opinion. I'm not going to tell you what to do for this, but maybe something like they gain a +2 to their AC while raging.
• By fang and claw: For this feature, this could also be another option that could happen at level 3 as rage modifier, but this does not belong as a 6th level feature for a barbarian. This subclass feature tier is usually meant for a defensive or utility subclass feature (ex. battlerager gains temp hit points, berserker gains two condition immunities). Maybe this could be a feature where they gain a resistance to the damage type that use for their breath weapon.
• Dragon re-born: Basically just what I said with the monarch of the sky, these two feature should be combined and possibly re-balanced to match combining these features.