Why do you need to revive Genesis, if you're already skipping Apoc, Alliance and Interceptor?
You already made an effort to create your own backstory with that document, and IIRC, the reason why XCOM 6 was called Genesis is because if the player won the game, Earth would be restored to a pristine state because of captured alien terraforming technology, after the ecological mayhem caused by TFTD that led to the construction of the megacities in Apoc.
Also, that document is good for fanfiction but most of the historical events described are irrelevant to a player - someone might read it out of curiosity but it's still 20 pages - since they will serve no propose for gameplay. For you to have an idea, this is the current background for the Tech-Comm mod:
INTRODUCTION
It is 2027, nearly thirty years after Judgment Day.
The war against the machines has entered a dangerous phase: in 2025 the resistance attacked Cheyenne Mountain in a gamble to destroy Skynet's Core and end the war. After months of preparation and gathering the necessary forces, the resistance forces were able to overcome the complex's massive defenses but the A.I. had long predicted such a move and had already relocated itself to a currently unknown location prior to the attack.
Although successful in taking the complex despite the heavy casualties, the Resistance's victory become a phyrric one after the battle: Skynet immediately launched a counter-offensive worldwide to terminate key resistance units and bases, taking advantage that many had redeployed for the Cheyenne Mountain offensive.
For the past year Skynet has reduced its activities, allowing the Resistance groups to recover but reports indicate that the machines have been developing new technologies and mass producing new Terminator units and Hunter-Killer craft, and getting ready for a new offensive destined to crush the Resistance and drive humanity into extinction.
However, at the insistence of John Connor, one of the Resistance's most successful leaders, the human factions have decided to combine their efforts into creating an elite unit charged with researching the machines' technology in order to defeat Skynet's legions and win the war for human survival.
And nothing else is required, because the essential of the story will be told through gameplay - and this is what you should be focusing, on what exactly will take place during the game and start implementing it. Otherwise you'll simply be writing design docs with no end.
Just my 2c - good work on your project