I remember a bit about the history of Rabban Gamliel. he's the one who felt learning was only for the elite. I remember when he was fired from his job as Nassi, and all the problems taht arose. Still, I find it hard to reconcile the fact that the Shekel process was to make everyone equal in the upkeep of the Temple, and the fact that Rabban G' made sure his Shekel was used for a Korban, and not for maintaining the walls. I was told as an answer that just like a Rebbe, who gets people asking him to pray on their behalf, says he doesn't understand why people bother him, he's just a simple Jew, and yet people find comfort and solace in his prayers, so too the people felt comforted knowing that this Shekel was used for a Korban. I'm sorry, but it just doesn't make sense!
Concerning your difficulty in understanding the actions of the members of Raban Gamliel's house, they were not trying to get a bigger or more important share in the Korbanos, or to fulfill the Mitzvah better than others. Rather, they were trying to fulfill the Mitzvah in a zealous manner (with "Zrizus"). The Tiklin Chadetin explains that this is the reason the Gemara proceeds to discuss the saying of Rebbi Pinchas ben Yair, who discusses this characteristic of zealousness.
Rabeinu Shlomo Siriliyo writes that certainly, the members of the house of Raban Gamliel knew that whatever money was selected for the Korbanos was selected on behalf of all the Jewish people. They merely wanted it to be their Shekel through which others had a share in the Korbanos.
(Concerning your understanding of Raban Gamliel's conduct as related in Maseches Berachos, the commentators explain that Raban Gamliel did not feel that learning was only for the elite, but rather he maintained that before teaching Torah to a student, one must determine the nature of that student to ensure that his motivations are pure.)
Be well!