The series is set in a different timeline, yes.
Not exactly. When the Pillar of Autumn retreated after the battle of Reach, they believed that thr Master Chief was the only surviving active spartan, as the rest of them were on Reach*, or on the Autumn but clinically dead (you can actually see a display of her in cryo in the remastered version of CE).
As it happened a small number of Spartans did actually survive Reach and were later rescued, but no one not on Reach knew they were alive during the events of the first game, and were present on Earth during the events of Halo 2, but not 3, for complicated reasons.
Irritatingly, there was one other Spartan still alive and on Earth as of Halo 2: Maria 062, who retired from the military to raise a family and who's never been followed up on after her one appearance. And a series following her, trying to survive and protect her family on a Covenant occupied Earth, would be such a great and simple story to tell.
*Save for the 3 Spartans on Grey Team, who were out of contact with the UNSC behind enemy lines.