This is such a good idea. I've been wanting dragons to have a much bigger role in the series for years, being more than protectors of a province. Rather I'm still hoping for land-based dragons; that is, those who patrol the overworld and will attack if they notice Link. Might sound a bit like Skyrim, but then The Elder Scrolls series seems to take many things from LoZ, adding its own twist to each aspect of course. So I don't see why Zelda can't do the same and borrow dragons, for one thing, as an adversary pulling off its own interpretation.
Considering that, the participation of dragons in the Zelda franchise can take on many forms. They can be standard enemies, who appear regularly throughout the game and even grow more difficult as the story progresses, like the Bokoblins in SS. In Twilight Princess there were these huge vulture-like birds which pursued Link when spotted. I can see dragon enemies doing this same thing. Secondly Link could have a dragon that he uses to navigate certain terrains (such as breaching a large, bridgeless chasm). Just think back to the Shadow Kargorak, also from TP, which we had to ride in order to reach the higher grounds belonging to the Zoras. Of course riding the dragon wouldn't be so narrow, but more like embarking on a Loftwing through the open Sky.
Lastly the role I'd most of all want to see a dragon take part in is that of an overworld boss. I know that it could virtually be a repeat of the battle with Argorak, but this would have a different caliber altogether. It would actually be more like the confrontation with Ganon Beast in Ocarina of Time: you could move around freely and try to get around to the weak spot (just say it's under the beast's stomach), and attack when it's open. Meanwhile the dragon will be defending by blowing fire, swiping out its claws, and whipping its tail around, in sequence. It can even perform aerial assaults by diving and hacking, in which case Link would have to mind where it would be attacking. The end result would feel like an genuine dragon fight that would, in my opinion, be a fitting addition to Zelda.