Well, it could vary a lot. In some fanfics, Bardock might be reborn with new powers or a different destiny. Maybe he gets to change the future of the Saiyan race. But it really depends on the author's creativity.
There are no such inappropriate 'Straight Shota' stories in the official Dragon Ball Z canon. Dragon Ball Z mainly focuses on epic battles, power-ups, and the adventures of Goku and his friends to protect the Earth from various villains.
One alternative is to focus on character backstories. You could write about the early days of Piccolo, how he came to be the way he is. Another option is to do crossovers with other non - sexual universes. Maybe a story where the Dragon Ball Z characters meet characters from a magical anime world and they have to work together to solve a problem. Also, you can write about the relationships between the characters in a platonic way, like the deepening of the friendship between Gohan and Krillin.
One interesting plot could be a new villain emerging from a long - forgotten dimension. This villain has powers that are a mix of different techniques from all the universes in Dragon Ball. He starts causing chaos by stealing the energy of strong fighters to open a portal to an even more powerful realm.
Vegeta is also very popular. His pride and his journey from a villain - like character to a hero - like figure gives a lot of room for creative writing. Fanfiction writers often explore his relationship with his family and how he balances his Saiyan pride with his new - found sense of belonging on Earth.
Obviously, Bardock is one of the main characters since it's about his rebirth. Goku might also be involved as Bardock is his father. Vegeta could potentially show up too, as he is an important Saiyan in the Dragon Ball Z universe.
One interesting crossover could be a battle between Hulk's brute strength and Goku's Super Saiyan power. Maybe in the fanfic, they start off as rivals, with Hulk misunderstanding Goku as a threat to Earth. Their fight would be epic, with Hulk's limitless rage against Goku's increasing power levels. It could also explore how their different universes' power systems interact.
The full story of Dragon Ball Z is about Goku and his friends protecting the Earth from various powerful villains. It starts with Raditz arriving on Earth, revealing Goku's Saiyan heritage. Then there are battles against the Saiyans like Vegeta and Nappa. Later, the threat of Frieza emerges on Namek. Goku transforms into a Super Saiyan for the first time to defeat him. There are also the Androids and Cell arcs which bring new challenges.
A Kryptonian in the Dragon Ball Z universe could potentially be a very powerful addition. Their natural abilities like super strength, flight, and heat vision could make them a force to be reckoned with. However, they would also face challenges as the Dragon Ball Z world has its own power scale with ki-based powers. They might have to adapt to the different fighting styles and power sources. Maybe they could learn to use ki in addition to their Kryptonian powers, or they could find a way to combine the two in unique ways.