Summary: Bardock's son Kakarot is the prince of the Saiyan empire.
Comments: This too is actually based on a previous what-if column, but it's better updated to include a tribute to Malik Studios' Dragonball New Age by having Vegeta's son be Raditz, a reference to the character of Rigor. Basically, this story is what it would be like if Kakarot were the prince of the Saiyans, so the story revolves mainly around these characters.
e. King Bardock
I have it so that the Galactic Trade organization conquers Earth thanks to Raditz, so the main emphasis is on the Saiyans; I usually don't like how events in what-if fanfics solely revolve around switching events, so I don't have it so that Vegeta gains the ability to read the future. Instead, I'm going to have him become a character in the series after the destruction of Planet Broly; that, and why not have Bardock keep his ability to see visions of the future? That honestly makes it a better story.
Of course, this would revolve mainly around the intergalactic saga as Kakarot is slated to defeat Frieza. It starts when Kakarot and his fellow characters are adults, so we get a look at the difficulty that the Saiyans have in trying to restart their existence. For example the Galactic Federation Earth is where the Saiyans are trying to build family lives, but as a result of finding themselves the last ones alive, we would explore their psychic baggage. I would possibly have the catalyst for the battle against Frieza be when Vegeta gets killed.
As I said in the what-if column, I think this would be a telling of a Dragonball story that relies on one setting; space. I would want to tell this as if I had the chance to create a new Dragonball series based entirely on the what-if. I look at Dragonball Zero as a very respectable tale of Raditz's character and I definitely credit that as an inspiration, somewhere down the road. I wish whoever is creating it the best of luck! Having worked on an animated series, I can understand the amount of pressure that's needed to finish a good series!
I think the post-Frieza villain I would love to have is a bastard son Vegeta Jr. He would be the only other being in the universe to go Super Saiyan and as such, it would be the best fight in the whole series and end with the ultimate cliffhanger!