No, that's not the problem. The problem is that Dragonwrought does not technically require the dragonblood subtype. It requires you to specifically be a kobold. The fact that kobolds have the subtype doesn't really matter. I could see a DM allowing it, but it isn't really RAW.
It is RAW+logic, namelythat it requires dragonblood through requiring Kobold, it even says you lose the dragonblood subtype, not lose it if you have it.
No. It requires you to be a kobold. It is understood, by reading the Kobold Racial Traits entry under their chapter in the same book, that kobolds have the dragonblood subtype. Dragonwrought, itself knowing full well that normal kobolds are humanoids with the dragonblood subtype, stipulates that by taking the feat at 1st level, your racial trait swaps out the subtype and instead of even writing it down on your character sheet, you write "Dragon" as your type. And while I said before that fluff is fluff, it is still generally understood by reading the dragonwrought section of the racial writeup, that not only are dragonwrought kobolds born that way, their unique nature is generally apparent even before hatching.
Saying that you are "reborn" when you become a Dragonborn means nothing, other than your character
is a Dragonborn. Your character gives up being its old race, generally leaving behind its old identity and life, and steps into a new one, that of the dragonborn.
You can't try to use fluff to explain why you're disregarding a different set of fluff.
Now, since Bahamut doesn't exist in Arhosa, the whole serving him bit doesn't apply, but Dragonborn could be explained several other ways. Mostly, I just don't think it's a good idea to try and justify combining dragonwrought and dragonborn. They are two separate, unique racial qualities, both very dragon-flavored and good on their own.
The best way to be as dragon-infused as possible is to emulate them as much as possible. Play a druid aiming for Dragon Wild Shape, or a dragonfire adept or dragon shaman who thinks Dragons are pretty awesome, and wants to be like them (for instance, I'm currently playing my own class fix of the Dragon Shaman, trying to playtest it to see how it stacks up and where it needs tweaking. You can see it in my Sig), including by choosing a dragon type and roleplaying its unique personality quirks. There are plenty of ways to come at this issue.