Not true. That is the same belief that the old nords had, not the current ones. Titus Mede proved that.
My Dragonborn is a High elf. How would that work? Or an Argonian?
Sure, having the Dragonblood means something, but with all of the other variables, It'd be implausable and impractical.
Implausable, because there is no way that the nords would accept an Argonian to rule their province, plus they'd want way more leadership experience than what the Dragonborn currently has. (Which is why Jarls are the only ones up for the position.)
Impractical, because even if Alduin is gone, the dragon threat is still at large and gameplay would really lessen the credability of the position. For example, you don't want to simply be signing documents forevermore and if you went off adventuring it'd be hard to believe, at least not without the entire 2nd legion.