Indeed Goku is not a chosen one, He simply worked hardest at it and never did it for a selfish reason which is why he surpassed Vegeta, who did do it for a selfish reason time and time again
Gohan was considered special, for another reason. Because he is a half saiyan and half saiyans seem to have an easier time going super saiyan, Because most saiyans lack emotions beyond a superficial level (Goku shows a lot of emotion, but not deep ones.) Goku did get angry and upset, as shown with the death of Android 8 which triggered a huge power boost. Same goes for Vegeta where anger seems to be the only emotion he really has. It is when they trigger their polar opposite emotion that they became super saiyans.
Goku lost all control to anger = Become a super saiyan
Vegeta giving up and becoming sad = becoming a super saiyan
Bardock actually caring about others = becoming a super saiyan
Notice how this is their opposite emotion. Goku as a goof, Vegeta as an angry person, and Bardock as a pretty cold person.
For the half saiyans, Gohan, Trunks, and Goten. They had human emotions they were not stuck in their primary emotion (goku being goofy and vegeta being serious) Which is why Gohan was able to achieve super saiyan 2.
Gohan simply had to learn to let go of his control of his emotions.
Also notice how the half saiyans have a larger range of emotions than their full counter parts.
Trunks(future) did not fall into this category because he did not train with other super saiyans early on and the concept likly never arose for his time line at least. Trunks likely became a super saiyan because he wanted to make his father proud of him, A trait very common among younger children.
Goten likely became a super saiyan out of being lonely, He did not grow up with his father and his brother eventually went to school leaving him alone, another common childhood trait of some form of abandonment although he still had Chi Chi, most children in 1 parent homes do show that they feel that missing parent should be there.
Pan did not become a super saiyan because of time constraints, although her art work as a super saiyan is used in the online game.
Goku is very unique with not being a chose hero with nothing actually special about them. Think of all other heros especially in Anime, Naruto had a tails beast, Ichigo is part death god, etc
---------- Post added 2012-08-30 at 08:52 PM ----------
Also none of the half saiyans really had to become their polar opposites other than gohan, even then its not really a polar opposite if its his anger issues that arose constantly.