My initial theory, which I now am quite certain isn't correct because some things don't really fit, but still is quite fun is the following:
What if that device was originally designed by an Ascian and his or her intention was for it to be kind of a failsafe device: In case everything else fails, and all the Ascians are killed, this relic will remain and humanity alone will accidentally do the rejoining by themselves, due to them being flawed and always looking for more source of power/greed.

This would be extremely interesting because it would require a very smart Ascian to know that there's still a way to win even if they're all dead.

Of course, there are many holes in this theory. First of all, if the device has that kind of power, why wouldn't the Ascians have used it themselves? Also, even though the reflections get rejoined, it doesn't look like the people's souls do. (Unless the people that get born in Heritage Found have a denser soul?)

The second theory that I had - especially when it was emphasized how this power came from the prayers and deep desires of the Lalas from the Source to escape - is that this is some kind of Eureka device, that has the power to do pretty much whatever it is that's required. It seems like there's another condition that needs to be met however, because almost nobody was capable of actually using it.

Finally, I'd like to bring up the fact that the Ascians never got ahold of the artifact. Did they simply not know about it? Did one of them try but was ultimately killed like with Eureka?