I'm going to go into a bit of FF14 metaphysics to try and explain this, and I hope I understood the question right else this might just be an interesting tirade xD

So the way soul, mind, and body seem to work: the soul is a being's power--it is the literal power of life that enables everything else to function as life as we know it. Elpis touches on this a little bit: that a being without a soul would be seriously limited in its ability to form complicated thoughts and beliefs. Likewise, Dawntrail makes it very clear that the quality of aether in a soul (or even the number of souls!) can drastically increase one's power--limited only by the body (I'm not going to get into Feral Souls right now, that's a whole different can of worms).

Think of the soul like a battery for the body, and a canvas for the mind. What the Endless do is take the etchings on the soul and preserves them. With this, the likeness of a person is infinitely maintained. However, these--essentially--serve merely as instructions. On their own, they're completely useless without a soul to provide that life force to keep the being alive. Theoretically, one soul should do the trick--after all, that's what happens naturally.

However, the Endless are truly that--Endless. And sundered souls are not. They don't have enough aether to be immortal like an ancient, and the only real solution is to get more. Without the power source of aether from souls stored (presumably) in these terminals, the memories cannot sustain themselves, no more than we can conjure a departed loved one into existence by remembering hard enough. We don't have that level of power.

I imagine what's happening is that the terminals are storing the soul aether harvested and the memories are imprinted onto them, like they would be on a natural soul, but on a far larger, multi soul scale. This is kind of touched on in one of the earlier quests, when with the either shortage, Endless had to exist on Living Memory in "shifts", taking turns, because there wasn't enough aether available to manifest them all.

As for Erenville seeing his mother again. This...is tricky. I always took this as an optimistic assumption, but all I can assume is that once released from the cycle, memories will return to the lifestream, same as souls. What we don't know is what happens to these individual memories, or if they will simply disperse, as even a natural soul tends to, eventually, be cleansed of most memories, and what's left is little more than faded marks on a reincarnated soul. For any of this to be true, though, the Terminals would almost have to act like RAM, where as soon as power is cut off from it, all the data is lost--and in this case, allegedly released. But this last part is all conjecture, right now. It's not explained in great detail compared to the rest.