Braska married for love, despite going against social conventions. He also chose the two most unlikely guardians: a drunken man claiming to be from Zanarkand and a warrior monk fallen from favor. No one thought Braska could defeat Sin. No one believed in his abilities. No one took him seriously.
Boy did he prove them wrong.
Braska has the same dumb-as-a-rock determination inherited by his daughter Yuna. He ardently believes in what he's doing. He's calm, regal, and kind. He's more than willing to die for what he believes is right--so he does. Die, that is.
"Let's show them they're wrong. A fallen summoner, a man from Zanarkand, and a warrior monk doomed to obscurity for refusing the hand of the priest's daughter. What a delightful irony it would be if we defeated Sin!"
Braska inspires in Auron a deep dedication and respect. Like a samurai for his shogun, Auron would die to protect Braska. Funny, in a way he kind of does....
Did Auron want the place at Braska's side, to be the Final Aeon? While Auron may have envied Jecht this role, I think his devotion went a little further. Jecht wanted to give Braska what he wanted: a chance to defeat Sin, even if it was only temporary. Auron wanted to give Braska what he needed: a chance to see his daughter and his world grow up.
Had Braska followed Auron's advice, things would certainly be different. But Braska was determined, and Auron would do anything for him...except go against his wishes.
Braska seems almost like a mentor figure for Auron: calm when Auron is angry, kind when Auron is blunt. So many years later, Auron is doing for Yuna and her guardians what Braska did for him--expressing a voice of reason, taking care of details, being there for the others before they even realize they need someone. Whatever Auron offers as a friend, as someone to be trusted and depended upon, he gets from Braska.