I like Nightwing, but Dick could have become his own man as Robin too I think. Dick's character will always be tied to Batman in some way regardless if he is Robin or not, but Robin could have evolved over time with Dick's character like Nightwing did. It's just Robin is a bigger brand than Nightwing. So Dick giving that up for Nightwing will always lower his level of influence since Robin is one of the few iconic identities in comics. Plus it is an identity Dick created himself which to me makes it more personal instead of taking it from someone or something else like he did with Nightwing. Which is a name he has very loose ties to these days with him not really being involved much with Superman anymore.
But Dick was also very lucky the Nightwing name stuck. I don't want to discount that. Since many times when a character loses an iconic identity they struggle to find a new one. Donna has been suffering from that for a long time. Going from Wonder Girl, Darkstar, or Troia. It's the same as Tim now going from Robin, Red Robin, and recently Drake. It's always a gamble when it comes to these name changes. Dick could have very easily fallen into that problem.
Also Robin is very unique in its name for a sidekick. Since it's name is completely independent and also doesn't have age restrictions on it. It isn't like Batgirl, Kid Flash, Superboy, Aqualad, and so on where the names limit the character from growing older or growing independent from their mentor. Plus Robin doesn't wear a Batman symbol, unlike most of the Batman family, and instead has his own R symbol. It all wasn't planned out when they created him, but Robin ended up being a very well designed identity, imo. The only issue is that they didn't give him pants sooner, lol.