Coming off from Namek, this whole situation is even odder. Vegeta is blatantly evil throughout the entire Saiyan and Frieza Saga. You have to remember that Vegeta since he was a child was raised to be a space pirate, so the most logical next step for him would be something akin to taking over whatever left is of Frieza's empire or forming his own gang similar to what Turles did. I'm not saying this fanfiction scenario is necessarily what should've happened, just something you would expect from the prideful prince of Saiyans. Instead, however, we see him settle on Earth.
(In the anime, he goes searching for Goku in space, and once he settles we get a bit of brief character development between him and Bulma. Neither happen in the manga, and because of this I just want to make it clear that I'll be disregarding both of those character arcs in this theory.)
There isn't much I can say about how much out of character it is for Vegeta to be on Earth, since everyone who has watched the series has already realized this and/or just doesn't care and wants to see Vegeta continue to kick ass. I have a different view on the subject.
The theory is basically this: Vegeta stayed on Earth to repopulate the Saiyan race, and it worked.
There are other reasons one can speculate based off the series itself, like the fact that Vegeta obsessively wants to kill/fight Goku and could be just staying there awaiting his return like everyone else. That is also a logical point and is something a lot of people I think could agree on. Not denying that aspect of his character, but looking at the outcome and the development of the series itself points to some interesting signs:
- The first encounter with Gohan proved to Vegeta that both Saiyans and Humans are sexually compatible with each other. Not only that, but they're stronger than the average Saiyan, even where they are able to achieve SSJ form at extremely young ages that only a true, almost god-like seasoned warrior could obtain. On a scientific level, this is extremely rare but not surprising considering how similar Saiyans are to Humans.
- By Namek, Vegeta consistently allowed Gohan and his friends to live. While he had evil intentions, this was the very beginning of him getting on their good side. Why not just kill them with ease to get them out of his way? (He had several chances to do so!) Maybe Vegeta garnered a deep respect for the Earthlings after his defeat.
- Vegeta married into a family that is quite possibly the most successful and richest on Earth. Bulma is one of the most intelligent people on Earth, and the result was essentially a fusion of Vegeta's strength and Bulma's smarts in Trunks. Really, Vegeta couldn't have asked for a better combination. (Of course he wouldn't marry into the King's family...he's a dog. Probably not sexually compatible or viable.)
Did Toriyama put much though into this? Probably not. But I find it hard to believe that Vegeta would be just as oblivious as Goku is to his existence on Earth. Despite Vegeta's straight up incompetence at times and failures, he's still fairly intelligent and consistently aware of his place in the world on the average basis. I think it's highly probable he saw how successful Gohan was, and in result he tried to go one step ahead and fathered Trunks. Along with Goku's family lineage, his family lineage too will exist on Earth.
Does this mean necessarily he stayed on Earth to repopulate it with Saiyans? Well that's hard to say that is 100% grounded in reality, but it's something interesting to think about considering Vegeta has apparently had evil intentions pretty much the entire manga till around the Buu Saga. Though after Namek it's never made clear other than him wanting to get more powerful than Goku.
However one thing may be clear is that whether Vegeta or Goku intended it or not, it still worked, if you take account of Dragonball Online's backstory. Though just to make it clear before people wonder how this happened over simply the course of 200+ years, the human fighters that are part Saiyan are from an unknown point in the future to help Trunks save the past. So such a major course of human evolution definitely took time, even though it's ambiguous, all of the human characters in the game are part Saiyan which is quite extraordinary. Earth, essentially, became the new home of the Saiyans and it's all because both Goku and Vegeta lived there.
Though whether you take that bit canon or not, both Goku and Vegeta produced extraordinary children that have surpassed both of them. That is, by all means, the future of the Saiyan race whether it was intentional or not.