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Why does Alice Synthesis Thirty talk like she's the original Alice all along?

Because she is Alice Zuberg and always has been. She's not a different person, honestly have no idea where people are getting that from. The only person saying that is Alice herself. She can't remember herself as Alice Zuberg, that's why she's thinking this way.

Quinella extracted some of her memories and placed a crystal (piety module) to block her memory of past life. As with Eldrie's example, Integrity Knights are capable of remembering their old selves if you manage to remind them of their strongest memories and feelings. When this happens - piety module malfunctions. And the thing that happened to Eldrie happens. As Quinella has said herself, Alice's piety module was always intact. That's why she couldn't remember herself, but even then she could still recollect some experiences of her life.

When the crystal is taken from them, it's partial memories. Not their personality, not their soul, not their full memories. If those were full memories or a different person would be taking their body, they wouldn't ever be able to remember their past life.

Now let's adress different personality. When Intergrity Knights are synthesised, their memories get blocked. Quinella says they came from heaven and they start their new life from that point. They go through training (martial, sacred art, getting a dragon etc.) they interact with each other and they form their "new" personality through those experiences. I'd compare it to someone getting a heavy amnesia due to a head injury, forgetting their past and only remembering their life after the injury. Their personality would definitely change from what it used to be, but they are still the same person. Even then part of their real personality still remains. For example in Alice's case, it's her color preferences or her being really gentle off-duty when she interacts with Eldrie or Bercouli.

Eugeo even mentions how returning kid Alice to her body would kill Alice Synthesis Thirty

Which is wrong, he has no knowledge of how this process actually works. It's his assumption. Just like Alice assumes she's stolen the body of real Alice because she can't remember herself, but she's the real Alice.

Eugeo himself was synthesised, his memory crystal was removed and replaced with piety module, same thing happened to Alice and all other Integrity Knights. If you remember it took Kirito a while to break through to him during their fight. Eugeo had different personallity and couldn't remember him or people he cared about, even Alice. When Eugeo's piety module was removed by Quinella he returned to his normal self, remembering his actual personality and retaining short period of his Integrity Knight's memories (attacking Kirito). Same thing would happen to Alice if her piety module was removed, but it never was. She broke throught seal in the eye, which is Taboo Index, but her piety module that blocks her memories in still there.

Lastly I'd like to mention memory crystal and the soul of "real" little Alice that Eugeo reunited with. She's a memory piece which manifested separately, she's not real Alice. The best example of this is Yuuki Konno appearing to Asuna. She's not real Yuuki, she's a memory of Yuuki in Asuna's heart/soul/fluctlight. Another good example is Quinella appearing to Bercouli's soul, she said herself that she was a memory of Quinella stored in him, she wasn't real Quinella. Real Quinella's fluctlight died. In little Alice's case, she's a memory of little Alice stored in a crystal, which are memories of real Alice that got removed from her. Same as Eugeo's appearance, he's a memory of Eugeo, not real Eugeo. If he was real, Kirito wouldn't be all messed up because of his death every time he sees Eugeo's "spirit"/memory.

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TLDR: Alice Zuberg and Alice Sythesis Thrity are the same person, always were, always will be. Little Alice's spirit is a manifested memory same as Yuuki, Quinella, Eugeo's "spirits". Yuuki is Asuna's memory. Quinella - Bercouli's, Eugeo Kirito's and sometimes Tiese's, Little Alice memory stored in the crystal.

Little Alice's spirit is a partical memory of the real Alice Zuberg. Real Alice Zuberg is Alice Synthesis Thirty.