They serve a role if its important to the author. However I feel that Batmans progression was best realized without someone pointing it out. The character has become stagnet by making it a soap opera with unreasonable amounts of back up that take away from his insanity. For me the draw to Batman was that he was a hero that could easily be a super-villain. His motives, his actions all border on the insane. The side-kick I see it as just a vent. I believe one comic book writer wrote that. It was along the lines "Robin helped it so all the Batman's comics were just thought bubbles".
But Batman is still human in the comics. And no, batman is definitely no borderline villain. Maybe that's how you choose to view him, but it doesnt make it correct. In fact he is the one rubbing other people's mistakes in their faces as if he is infallible. He uses violence and extreme methods but he is no borderline villain, ergo he is no Rorschach.
Because they argue, doesn't make them different.
Bruce: Dark, sully, brooding, miserable, extremely focused, obsessive. Expert martial artist, detective and scientist. The goddamn Batman.
Dick: Lighter, focused on the mission but also having fun, enjoying life, being optimistic. Ladiesman. Expert acrobat and martial artist.
Jason: Angry, miserable, obnoxious, uncontrollable. Expert at nothing besides coming back from the dead after having his head bludgeoned with a crowbar.
Tim: Focused, a character that has dark and light in balance. He is darker than Dick but lighter than Bruce. Expert detective with Bruce himself stating that one day he would outdo him.
Damian: Son of a *****. Literally and metaphorically. Characterised by great anger, lack of manners and morality. Skilled fighter.
So, how are they same?
Robin as I have seen him is stale and contributes nothing.
Judging by this quote and by the previous one, i have to assume that you havent read enough Robin stories. How does he not contribute when Dick helped Bruce lighten up and find his balance? When recently Bruce looked back at his early years and was very fond of the fun adventures he had with Robin. He also took a new role, that of a father and mentor. Then when Dick left him his loneliness made him take in Jason, even though he later admits that Jason wasnt ready.
Also, after Jason's death Bruce has sworn never to take another kid in, but Tim forced him to do it because he felt that Batman needs a Robin in order to keep his balance. I suppose that those issues elaborated more on why he does need a Robin.
So how do the Robins not contribute anything?
Also he does have a son now, thanks to the ridiculous stories that are being told (which are addressed later).
Damian is his biological son, but he hardly knows him. Dick is his true son (and later so is Tim) who he has adopted and considers his heirs and children.
The problem I see, is that Robin isn't a son and it is never addressed. He is just another vigilante.
And now you are forcing me to assume that you ve read no robin issues.
1) Bruce adopted Dick. It was rough at first but they eventually developed a father-son relationship.
2) Tim's mother was still alive (IIRC) so he didnt need adopting, but at some point Bruce adopts him to show him his love (i assume that both his parents were dead at the time). Bruce said something like: "i would be honoured if you accepted" (to be adopted) and Tim said "i know, i love you too Bruce", while accepting.
3) Dick is treated by everyone in the DCU as Batman's son. When he is around Clark and Diana you always get the feeling that they are his uncle and ant.
4) Dick and Tim, now both adopted by Bruce are heirs to his fortune. Recently they were following a lead about a rich guy so they flew to Monaco acting as the little Waynes that they are to infiltrate the rich society there.
5) At some point, Bruce became too obsessive and was about to denounce his Wayne persona and become Batman full time. Dick (he wasnt even a kid, he was Nightwing at the time) burst into tears saying "You cant do that. Where is the man that raised me? He never existed?"