With Sword Art Online's sprawling Alicization arc having concluded, it's the perfect time to look back on some interesting facts and speculate about the monumental story. Alice's incredible journey from a young girl studying sacred arts in the VR-simulated Underworld to a warrior fighting in a full-scale international war and ultimately entering the real world felt like fate much of the time. However, it also hinged on some unlikely coincidences, pure luck and, most of all, Kirito's intervention.
Rath's goal in creating the Underworld was to create a bottom-up artificial intelligence being capable of thinking for itself, adapting to various individual situations and even disobeying commands if or when necessary. If successful, this being would become a model for future real-world AI soldiers as well as improving Japan's defense technology. When Alice accidentally disobeyed the Taboo Index by crossing into the Dark Territory, she became the primary target for the ambitions of Rath as well as the Gabriel Miller-led American team looking to steal the tech for the National Security Agency. However, had Rath gotten hold of her as soon as they'd liked, they would likely have been disappointed with their findings.
Because Rath only had a general overview of the Underworld with certain 'alerts' in place, they could tell when one of the Underworld's residents broke the Taboo Index but not see specific details. They were unaware of Quinella/Administrator's autocratic rule and her horrific ritual for turning skilled fighters and lawbreakers into fully obedient Integrity Knights -- in many ways the exact opposite of what Rath wanted. So despite Alice deviating from the system -- even if by accident -- Administrator essentially removed her past memories and reprogrammed her mind. Had Rath gotten hold of her lightcube then, she would have been a powerful fighter, but not the independent thinker and planner they were looking for.
Of course, Rath's original aim for putting Kirito in the Underworld -- a human unbound by the Underworld's laws and restrictions influencing those around him -- was crucial. It wasn't Alice Zuberg breaking the Taboo Index that made her the AI everyone wanted. Rather, the true breakthrough was Alice Synthesis Thirty breaking the Right Eye Seal and consciously choosing to act of her own accord. At the time, Kirito didn't know that Alice was the exact Underworlder whom Kikuoka and Rath were looking for. It was merely a lucky coincidence that his goal -- to convince Alice to join him and Eugeo in fighting Administrator's dictatorship -- aligned with Rath's goal of achieving fully autonomous AI.
Although Eugeo was the first to break the Right Eye Seal when he helped save Tiese and Ronie from being raped, the time acceleration between the real world and Underworld (1000x faster in the Underworld) meant Rath was totally unaware of his actions. Between Eugeo breaking the Seal and his later death in the battle against Administrator, only a few minutes would have passed in the real world. He had never broken the Taboo Index in any way before that, therefore Alice was the only one the two opposing real-world factions ever thought to pursue.
So, what would have happened if Rath or the invading American team had gotten hold of Alice Synthesis Thirty's lightcube before she broke through the Seal? It's almost certain the Americans would've continued their plan to bring her back to the United States, even if Alice wasn't yet fully autonomous. However, if Rath had reached Alice when they knew of her initial Index-breaking actions, they likely would have been confused at her still-obedient nature and sent her back to the Underworld to continue her life there. In their eyes, she wouldn't have been fully developed yet. On the other hand, young Alice Zuberg's mind subconsciously acted on its own when she wanted to help the injured Dark Knight. Perhaps Alice, regular girl or Integrity Knight, was always on the path to becoming a free-willed individual and just needed a boost from Kirito. Fate indeed then.