Hey there! This is Nathan from the PROJECT: Overdrive team.
The story definitely raises a lot of questions, so I should lead off by pointing out that the team wanted to make PROJECT an entirely new alternate fantasy.
Yi's visions (and I'll call them visions rather than flashbacks) are a pretty mysterious anomaly. The PROJECT armor was born from an effort to link technology and magic to empower the champions to channel the supernatural ability they already possess (and also to look totally badass). PROJECT is the collision of a lot of opposites: past and future, malice and humanity, magic and technology (to name a few). You smash all those things together inside someone's head with rivets and wires and sensors, and you're bound to dredge up some specters from the depths of their psyche.
You raise an excellent question about the intent of the man in the white coat who would appear to have a score to settle with Yi's forearm. He's a bit tougher to pin down as far as his motives go. His only crime was loving science too much!
As for the city, I really hope it was as epic as the team set out to make it. As a player I would love to grab a hovercar and explore that world for myself. As for having the city show up in the game I couldn't really say, but just like with Overdrive I'd jump at the chance to do more with PROJECT. For now, I really hope that you take the video, the skins, the site, all of it, as an invitation to draw your own conclusions and let your imagination run wild. There's a lot of creative space to fill and questions without clear answers, and that's by design. Clearly one video couldn't encompass everything, certain champions included. As a Leona player and an utterly incompetent Yasuo (AKA the one on your team), the conspicuousness of their absence was not lost on me nor on the rest of the team when the final cast was locked in. The most important thing is to keep telling us what you think. Your passion and your responses to the content we make is what compels us make more of it and to strive to make it better than what came before.
PS. I'm trying to remember where the team came down on putting easter eggs in the video, but my memories seems to be corrupted.