The position gained would have made a difference if the team got the penalty repealed after the race. Pushing anyway was the correct course of action, just in case.

He was trying to make up those 5 seconds lol
Futile? Maybe, but he gave it his all!

As opposed to just not trying to race at all?

Penalty reversals based on appeals or new information, penalties to drivers ahead, there's all kinds of reason to push to the very end.

He also have to sell himself. Worse case he's also learning, and the team is learning more about the car

Just for a racer's natural will to race I guess.... I suppose if he had got ahead with a couple of laps to go he could've pulled a second or two out and hope that the cars got backed up behind so he could squeeze 10th