John Carter (film)/WMG

Edgar Rice Burroughs does not write the Barsoom stories in the movieverse.
The first few Barsoom books had the Literary Agent Hypothesis where Burroughs was just publishing the memoirs of his uncle (who actually gave permission). Eventually the pretense was dropped. John Carter's body was said to lie dormant in a vault as his consciousness was in a duplicate body hopping around on Mars.

In the movie, unlike the books, the evil Therns have infiltrated Earth and John Carter's vault and the body inside now have to be guarded by Secret Keeper Burroughs. Assuming there's more than one Thern agent on Earth, why would Burroughs expose the Therns to the public and draw more of their attention to him?

There's nothing preventing him from writing Tarzan and everything else he did, though. Carter's parting advice to "go write a book" could have easily been an allusion to that instead of the Barsoom tales.
 * Of course, that might be why there's nothing in the original series about the Therns on Earth, or his uncle's vault: Edgar Rice Burroughs had to avoid attracting their attention.

The Therns aren't the only offworlders on Barsoom
The Black Martians, or First Born, aren't native to the red planet either in movie-verse. It's the only way their dynamic from the novels makes sense, as the movie Therns are so advanced that it's doubtful a native Barsoomian species could raid and manipulate them for centuries. Likely they've been playing out the same basic setup as depicted in The Gods of Mars across worlds for untold milennia, with Matai Shang not mentioning them in the movie because he doesn't want to admit weakness or fallibility in his people to either Sab Than or Carter- much less that they have enemies.

This also frees up the designers to remove some of the Unfortunate Implications of having a major group of villains be black-skinned Human Aliens by making them obviously nonhuman in appearance (the depiction of Issus- herself a First Born- in the Thark temple is just a vagule-female humanoid silhouette, which tells us nothing about what they really look like and may be pure artistic interpretation in-universe).

John Carter is an Earth-born Thern
In the original novels, John Carter is an immortal of uncertain age, with no explanation. In the movie, there are hints that such may be the case, albeit nothing explicit. He's also the only person in the film who ever makes use of Thern technology stolen from one, as opposed to given. And the Therns show specific interest in him, despite knowing well that Earth exists and has people on it. So, my theory: John Carter is actually a Thern ( or perhaps part-Thern ), probably the result of a rogue Thern operative on Earth some time in the past. He doesn't know this, but Matai Shang knows or suspects this, and it is for this reason that he is allowed to live.
 * Part Thern makes the most sense (he has hair, after all), but since the film explained one of the big mysteries about Carter (how he gets to Mars) it certainly seems likely that they'd explore the other (his longevity). Alternately...
 * Wouldn't a Thern having sex with a "lesser species" be like a human having sex with a cow? I know there's no indication in the film that they eat Red and Green Martians but still.
 * Well, in the books Phaidor was fairly open in her attraction to Carter, while Matai Shang is one of the many, many male villains with a thing for Dejah. It's also stated that Thuvia was thrown to the banths for refusing a high-ranking Thern's advances. Therns using "lesser beings" for purely sexual purposes would likely be considered mildly taboo, but that doesn't mean it doesn't happen.

The Therns' immortality comes from exposure to the Ninth Ray
Since it powers their technology and abilities, they're exposed to it constantly, and either deliberately or as a side-effect it extends their lives to the point that they're functionally immortal. This also explains Carter- his immortality is never mentioned early in the film, but it's clear at the end that he's not visibly aged since visitng Barsoom. Being transported by the Thern medallions may have given him enough of a "jolt" to at least extend his life by several decades...
 * In the books all Martians were effectively immortal thanks to their advanced medicine, the Therns just didn't have the "one-way trip down the river Iss when you reach a thousand" tradition.
 * In the film, though, the Therns are older than Barsoom, rather than being descendants of its original dominant race (or at least, says Matai Shang, who may have been lying).

The Therns aren't a race.
They're the collective consciousness of a godlike swarm of nanomachines that turned against their masters billions of years ago. Each individual we see is a pseudopod, or a scout unit, of the whole, disguised to appear as a human. Their fancy fractal weapons and tools are all extensions of the nanomachine matrix.

The Therns aren't offworlders.
They're the descendants of the White Martians who once ruled Barsoom, just like in the books. Their ridiculously advanced technology was either based on something their ancestors had but the Red and Green Martians have lost or something they invented since. Matai Shang was lying about their being offworld immortals purely to mess with Carter's head and try to make him despair. They are planning on moving to Earth (and are already scouting it out) because Barsoom is dying, but it'll be their first actual planet-hop.