It's a causal paradox, specifically a bootstrap paradox.
As I explained earlier, the only way to make heads or tails of it is to assume there is a "Zeroth Loop" that we cannot observe where, say, the Auri treasure hunters really did summon Alexander, and the Illuminati later independently "refurbished" him, leading to the events of the raid. The Illuminati shoot the Auri treasure hunters in their confusion, and Backrix loses his footing, his journal tumbling to the ground without Schrodinger's influence. Past Quickthinx gets ahold of it. We genuinely proved ourselves worthy to Alexander, and after her sacrifice he takes pity on Mide and Dayan, releasing them in the distant past. With the knowledge of Alexander, his tale is passed down in the Hotgo tribe as legend until near-past Mide and Dayan try to summon him - however the time loop is now in effect, meaning Alexander now has to exist as a fixed entity in time. To this end he creates Schrodinger / Shanoa, an automaton cat he acts through that will nudge events to keep the loop perpetual.
In other words, the "Zeroth Loop" is a genuinely alternate timeline that may no longer exist because Alexander is self-observing. Since it exists outside of our ability to observe, such as it is, it's anyone's guess what the "Zeroth Loop" really was like and by extension how and when Alexander first came into existence. (This is all highly hypothetical, mind.)
Except for the "Zeroth Loop," we're all just actors on the stage, unwittingly performing the same worthless comedy show for eternity. And even if we did know, we still couldn't perform anything but that same worthless comedy show. Exactly when or how Alexander was first summoned nobody knows for sure since the "Zeroth Loop" cannot be observed by us, but the events surrounding Alexander are the way it had - well, has, I guess - to be... Can't turn out any other way.
I'm probably getting way more involved than you asked for, and for that, I apologize.
Side note: the Echo is not time travel, it's just the ability to dive into others' souls and view their memories.
I'm not a fan of it in particular, but I do enjoy how it makes me think. Usually it's just "bad guys are doing bad things, go stop them!" Here I'm having to actually think... if only a little. The first BlazBlue game ran heavily on a similar setup, and Sonic 06 had a big one as a part of the story, so it's not really anything new to me.