Well, I had a very long answer typed out, but lost it due to internet bugging out on me :(
Basically, I agree with you. Fun, friendly and smaller-scale competitions would be awesome. But anything that relies on the normal AQ warring mechanics is going to result in an Igneus/Aerodu war while the rest watch from the sidelines.
As for the different Clans and their popularity, some of it has to do with how they've been portrayed in previous quests, but there's really been little to no character development for the Clan leaders. And the Clans themselves haven't been a relevant part of the game since the early days.
The biggest reason, imo, for clan popularity ties in to pop culture. This is just my opinion based on my own limited observations, but if I had to rank the AQ elements from most to least popular, based on their portrayal in pop culture, it would have to be:
1: Fire - awesomeness, destruction, power, strength
2: Air - awesomeness, ninjas, flying, speed, stealth
3: Darkness - stealth, edgy, ninjas, rebel
4: Light - heroes, good guys, happy, noble
5: Earth - strength, defense, loyalty, stubbornness
6: Ice - cool, unflappable, edgy, bad boy/girl
7: Lightning - speed, power, overshadowed by fire and wind
8: Water - peaceful, passive, healing, weak
I had explanations as to why I thought that way, but it would take too long to re-type all of that. I put a few descriptive words relating to why I think they are popular or non-popular, in case that looks confusing.