What we know about Zeke and the Homelanders: They have been living outside the Walls for a long time (long enough for Zeke to not even know if they speak the same language as the people within the Walls.
What we know about the people within the Walls: They have no memories of a time before the Titans because of the Reiss family's ability to manipulate their memories with the Coordinate.
What we know about Titan Shifters (and normal Titans): While in Titan form, the Shifter (or normal person with no control) don't age (as evidenced by Ymir's backstory).
My theory: The Homelanders are a group of people who were living at the start of the Titan "apocalypse". After the outbreak, they discovered the Titan Shifting ability, and used it to "camouflage" themselves, so they could live among the Titans. Eventually, they founded the "Homeland" a (relative) safe-haven for them to live away from the Titans (which is where Reiner, Annie, Bertolt, and Marcel were born.
Now, since they didn't live inside the Walls, they were not subjugated to the First King's mind-altering abilities, they remembered everything; not just the rise of the Titans, but of humanity as it was before the Titans emerged. They remember every war, every sin, every catastrophe, genocide.... EVERYTHING. And, as they live in isolation, they start to believe that humans don't deserve to live. The Titans are meant to wipe out the human race and end the "cursed history" that they have created themselves.
Flash forward to Reiner, Annie, Bertolt, and Marcel. They have been born and raised in the Homeland, and they've been raised to believe this philosophy: humans shouldn't be allowed to live anymore. As they have been educated to believe this since birth (rather than adopting the philosophy later in life like their elders), they are deemed the most fit to carry out the Homeland's plan to destroy the society within the Walls. They're willing to comply (some more willing than others), and embark on their mission.
However, after actually meeting and getting to know the humans within the Walls (via the 104th Trainees Squad), they begin to doubt, and even regret their actions. However, they remain steady in their conviction too destroy the human race (as they know about the human race's history before the Titans).
How does Zeke fit into all of this?: He's the mastermind, of course!
We don't know how the Titan apocalypse started, but we do know that Zeke can transform humans into mindless Titans. My theory? Zeke started it. ALL OF IT. At some point, 100 years ago (in continuity, of course), Zeke, who was unhappy with the state of the world, perfected something (a virus or something, I don't know) that began transforming humans into Titans. He also created the Titan Serum, in order to keep himself and his friends and family (anyone he deemed "worthy" of life) safe from the Titans.
After the onset of the Titans, Zeke helped his followers build the Homeland, where he acted as their leader, spreading his "humans don't deserve to live philosophy among them. However, he eventually disappears without a trace (which leads to Reiner and Bertolt's shocked expressions at seeing him during the Clash of the Titans arc; they were seeing the Zeke! The legend who founded the Homeland!)
The First King: You might be wondering why the First King is in this. Well, it's known that the First King possessed the powerful Coordinate, and the Titan shifter ability. How did he come into possession of these powers? Answer: He was originally a part of Zeke's group (I'd say he was even a close friend or relative of Zeke's), and as such, Zeke entrusted him with the Coordinate (which I would assume Zeke created in order to keep the members of his group in line, if they ever disagreed with any of his decisions). However, the First King eventually became disillusioned with Zeke's ideology and fled. Believing that, if humans were kept under control of the Coordinate, and were constantly kept in fear of the Titans, they would reform their ways, the First King created the Walls in order to keep what was left of humanity alive.
Obviously, Zeke did not take to kindly to being betrayed by such a close friend (or relative), and decided to hunt him down (explaining why he suddenly disappeared from the Homeland), but did not find the community created by the First King until recently (after the Homelanders, ironically).
So Zeke's not trying to reclaim the Coordinate because he needs it (he can already control the Titans, and he has no reason to erase the memories of the people within the Walls), he just wants it because he can't stand that it was taken away from him. And after he gets it, he and the rest of his troops from the Homeland will murder everyone within the Walls, and definitively "end" the "cursed history" that mankind has created for itself.