I think it's an interesting diversion to explore whether or not half-bred Saiyans have the same drive to fight. It's possible, but I'm sure it has a lot to do with their upbringing and individual personalities. Out of all the Saiyan-Human children, Gohan is the most engaged per say, but only because he's the oldest, has lived through the most tumultuous circumstances, and has been surrounded by martial arts his entire life. We'll have to wait and see how Goten and Trunks develop in the current material because they could stick to being fighters. It's still safe to say that Dragon Ball is Goku's story even now during Super, but definitively during the original run of the manga.
