The world can have exactly zero dragons in its entire history, and the absence of dragons would not have any effect whatsoever.

It's not as if they have any impact on the plot either, when all is said and done. Actually, come to that, if there was some sort of official day of choosing (and why wouldn't there have been?), why hasn't Galby kept that going? He's got eggs he wants to hatch, after all. Why wasn't he also having them ferried around the countryside, or better still holding a public annual event when hopefuls can come to the capital and try their luck? It's not as if the Varden could have done much to interfere with him and Shurikan right there.