I had a wall-o-text typed up, but the 3000 character limit hamstrung me. I'll just shorten it to this:
There are no signs in-game or in other material that the Ascians were the ones to trigger the cataclysm that brought down their civilization. We know from in-game text and other official materials that they actually did have enough power to stop the cataclysm themselves - or would have, if its very nature hadn't been the direct cause of their powers going out of control. The resulting loss of control after hearing that sound was that ultimately necessitated the first summoning of Zodiark as they could not fix the problem themselves. Granted it was their own runaway powers after the fact that turned the sound from "huh, wth is this" to "oh crap, the world is ending".
While I concede that Emet-Selch wanting to kill all the things is spitting in the face of those who gave their lives, I do not believe it would restart the cycle. The world is already healed, after all. They only needed Zodiark to help with restoring it because it had been rendered completely incapable of even supporting life by the time the cataclysm abated. Fully rejoined Ascians would be able to populate the Source with new life themselves since it would still be able to support life this time.
The many sapient creatures resulting from the sundering legitimately have not evolved at all in the way the Ascians want to see. Their aetheric and physical capabilities have grown only as a direct result of the rejoinings carried out so far. The creatures failed to evolve on their own, unfortunately. While more intelligent than in the past, their technological achievements always seem to be traced back to the Ascians. Garlemald is copying Allag, and Allag got the bulk of its technology through Ascian involvement.
There really wasn't conventional conflict in the ancient Ascian civilization. Their "proxy wars" were more like chess games played through mindless creations incapable of making decisions, let alone feeling anything resembling pain - or feeling anything at all for that matter. There doesn't appear to be anything in-game or other materials that would suggest that their civilization included the impoverished or that there people were physiologically capable of becoming ill in the conventional sense. Indications are instead that they had a truly Utopian society until everything went to pot.
The majority of wars in the FFXIV timeline actually weren't the result of Ascians. A lot of the time it's just the "human" races having their petty disputes, crossing their collective blades, and going their separate ways after spouting a few platitudes back and forth. While most known cataclysms have been the work of Ascians, there are a few of those that came about through the hands of mankind as well.