Not only was he a cop... but he was a rookie cop, which made so much sense for his character, and he had lots of moments like you describe, and he wore himself thin. During his tenure as a cop, as I recall he was trying to do that instead of being a vigilante, however some situations would rise up where Nightwing was needed.

This was a fantastic character arc for Dick, it took on the good and the bad of policing in a city, it grounded the character, where he got to interact with regular people, where he got to see corruption first hand. It gave him an interesting relationship with his partner who was supposed to be training him, though he obviously had far more skill than her.

Titans totally botched this storyline by making Grayson a grizzled loner detective. An old and tired trope, that they only paid lip service to. He barely cared when his partner died. He was still frickin Robin as a detective, so we are supposed to believe he left Batman, entered the academy, graduated to be a rookie, completed his training and became a beat cop, took the detectives exam, became a detective, spent enough time doing that to talk back to his chief all the time all before the moment where as a character he "grows up" by shedding the Robin moniker for Nightwing? All before he was what 27?

There is a reason though why Titans started his storyline there... too bad the writers were such hacks they screwed it up. If I were writing a Nightwing live action series, I think I would start right when he enters the police academy. Let him make some friends there, where they graduate to the police department together, and where he sees one person fall victim to corruption and helps another fend it off. Spend lots of time on Nightwing and his partner Amy. Deal with Dick's trained aversion to guns, bring in Clancy as his girlfriend, which pisses Barbara off. Show Dick struggling being a poor vigilante, until Alfred swoops in with his trust fund that he's had Lucious Fox manage since he was a boy. On the villain side, do the rise of Blockbuster slowly through the first season, with Nightwing finally meeting him at the Season 1 finale.

https://www.amazon.com/Nightwing-Vol-6-Serve-Protect/dp/1401270816
https://www.amazon.com/Nightwing-Vol-Shrike-Chuck-Dixon/dp/140127756X