A bunch of (undead, vampire) super-soldiers have to protect humanity (the white race) from hordes of invading aliens. The threat is everywhere and omnipresent, even though the actual baddies are underwhelming and you can readily slaughter tens of thousands of them. They're almost all uniformly evil, apparently just for the evulz. They're controlled by the all powerful yet unseen (((Darkness))) and the only thing protecting humanity is a giant white golfball. A white golf ball. Violence is the only way to interact with anything, and violence and death are goals unto themselves.
I keep trying to play this game and keep ending up frustrated by how empty, brutal, and fashy it is. Your characters, based on their actions, would be complete pyschopaths IRL, and the bad guy factions are all empty caricatures of evil evil baddies including Alien Invaders (Cabal), dirty illegal aliens who steal things (Fallen), Space Muslims (Scorn, Hive), and whatever the taken are (lazy writing so they can re-use assets).
Idk, I'm kind of done with it. Grinding for loot so you can grind for more loot is fun, but the story is so dumb and unselfconscious, and not even fun. Like with WH40k it's fascist and it's glorious because it revels in the excess and stupidity of fascism, it's all a huge fucking joke that's completely over the top. But Destiny plays it straight and it's so... dour and boring and unpleasant. This is the second time I've tried to play it and the second time i've given up in disgust.
There's nothing that conclusively proves this, we have no way to know for sure why the Traveler stayed. But everything else you cover is broadly correct. There is a lot of interesting lore in the game, but it's both hidden away most of the time and rarely used for the actual in-game storylines.