They did do this - on the 14th world, they immediately beaten up the WoLs who barely had any power of light, except the world got plunged into darkness and creates a void. The Void killed all the inhabitants and made it a dead zone. No life can live there, only voidsent (who are hungry thirsty Aether beings). That's when they realize they can't immediately destroy a world, but slowly push it towards oblivion. To speed things up and to ensure no world have an imbalance of light/dark to the point it creates "The Void", they have to slowly tip the scales, and then have the source deal with a similar problem for the main world to absorb the shard's aspected aether. That's why they sent a couple of ascians in each shard to handle the problem. Lahabrea was sent to deal with us - an unsundered Ascian no less. It's just that, our power of light was stronger than what lahabrea can imagine at the time.
Similar to the ShB plotline, the Ascians plunge the world of the First very close to Light, then have the main world - the source - absorb all the aspected light aether. You learn more about the Ascian's real plan if you complete all of the Role Quests and do the final Role Quest plotline - Ardbert's final party member that didn't die.
The only problem was that they didn't expect Ardbert and the gang to kill the Ascians on the first, which resulted in tipping the world into Light - and only through Minfilia's help were they able to stay the flood. With Minfilia stopping the flood, she basically brought the world back from the brink - which allowed the Ascians time to force the Source to light. It's just... we're been growing too strong and out of their control that we stopped their plans each time at the source for them to succeed. Only through ShB's 5.0 intro plot do we learn that they eventually resorted to killing us using a "deadly poison" instead.