True Crime: Streets of LA and True Crime:Streets of NY were basically GTA from the perspective of a cop. You had the same open world to mess around in and pick up missions or stop street crimes, but you were constrained by the law. Commandeering cars was possible but you flashed your badge to do so, and I remember in the sequel if you screwed up and ran over too many pedestrians etc then you were booted back to uniformed beat cop until you completed enough minor crimes to become a detective again.A legit police simulator. LSPDFR is nice and all but the modding it takes to get going is too much for me these days. Rockstar needs to make an offer to buy the devs or just work on their own project.
Not a very complete simulation, but the games were pretty well received from memory so it's a shame the series didn't continue. We might have ended up with the game of your dreams.