A tiny fighter is not going to be as powerful or as lethal as a capital ship, or capital ships won't exist. However, that's only looking at one side of things. Keep in mind the number game here. For the price of one capital ship you can build thousands of fighters. The reason they exist is that they can be the most economical approach to fitting their need. It may not make sense to build a huge capital ship that cost as much as a thousand fighters if 500 fighters together can do the particular job needed as well as a capital ship.
You see this with modern combined arms approach of the military. We have many different crafts of varying size and shape, each made to suit their need. A tiny frigate won't take out a destroyer, but it can prove more economical. In fact the old school uber-destroyers were pretty much done away with because smaller ships (okay, and more so aircraft) proved more flexible then putting all your eggs in one basket.
As to how that would factor in to your brother's and their little competition of importance, remember capital ships also have thousands of people. Sure one brother may be maning a capital ship that blows up an entire planet, but he is only 1/1000 of that crew. Can he honestly say that the ship wouldn't have blown up the planet without him? Can he claim credit for the deed as if he was single handedly responsible. Even assuming he had one of the most important positions on a bridge, as opposed to being ship cook, the fact is that he had an entire crew backing him up and, to a degree, he just passed orders down to others.
The person in the fighter was the sole person in the game. When he blew up the bomber that almost got in close enough to destroy the capital ship he, and he alone, was the one responsible for it. He can say, without doubt, that if he were not there the bomber would have made it through to the capital ship and destroyed it.
Thus while fighters may pack less punch, they may still get to be the super stars because they are they are out there making decisions on their own, living and dying solely on their own skill alone.
This could even set up a bit of a 'fight' over who did more. The capital ship brother says his ship did more damage, the fighter pointing out that he can take sole credit when he saved the day, who was more important? Is a small cog in a huge weapon more important then the sole cog driving a smaller one?
If you wanted to emphasis this sort of difference I would suggest having fighters be given extensive leeway in decision making, possible scouting far away or communications tend to be jammed during a fight, so he can show initiative to do something important when there was no one around to give him specific orders.