The Saiyan race would likely face more destruction and chaos. Without Bardock saving Vegeta, the leadership of the Saiyans might be in disarray. The power structure could crumble as Vegeta was an important figure. Enemies of the Saiyans could take advantage of this situation and launch more attacks. It could also lead to a loss of hope among the remaining Saiyans as one of their strongest leaders is gone.
Goku's storyline would be affected in several ways. For one, without Vegeta, Goku might not have the same intense rival that pushed him to become stronger. Also, the Saiyan lore that Vegeta could have passed on to Goku would be lost. This could change Goku's understanding of his heritage and his place in the universe.
A 'bardock vs king vegeta fanfic' could explore the backstories of both characters. Bardock's story as a Saiyan who starts to question the Frieza regime might contrast with King Vegeta's more traditional and perhaps self - serving rule. Their fight could be a symbol of the changing tides within the Saiyan community. It could also introduce new elements to the Dragon Ball lore, like different forms of Saiyan power that are unique to this fictional encounter.
Yes, there might be some out there. You can start by checking popular fanfiction websites like FanFiction.net or Archive of Our Own. Just search for 'bardock x king vegeta' on their search bars and you'll be presented with various fanfics related to this pairing.
In the fanfic, Bardock might find a connection between the Saiyan and Namekian races that was previously unknown. There could be an ancient prophecy on Namek that mentions the Saiyans, and Bardock's arrival could be seen as the start of the fulfillment of that prophecy. This could lead to a whole new set of adventures and alliances for him.
In the fanfic, Bardock arriving on Namek could lead to a change in the power dynamics. He could train with the Namekians and learn their unique techniques. This new knowledge combined with his Saiyan strength would make him an even more formidable opponent. And perhaps, his presence could inspire the Namekians to be more proactive in defending themselves against external threats.
Bardock may start as a typical Saiyan, focused mainly on fighting and power. But as he faces the threat to Planet Vegeta, he could start to develop a sense of responsibility. He realizes that there's more at stake than just his own glory in battle.
Another theme could be loyalty. Bardock might be loyal to the true potential of the Saiyan race while King Vegeta is loyal to the existing power structure. This difference in loyalty can create a great conflict in the fanfic. Also, there could be a theme of social change. As Bardock may represent the underclass Saiyans who want a change, and King Vegeta is the symbol of the old regime.
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In this fanfic, a key element could be Bardock's newfound power. Maybe he discovers a hidden ability within himself that gives him the strength to save the planet. Another element might be the relationship he has with other Saiyans. His comrades could play a role in either helping or hindering his mission.