True Crime was also total fucking shit. Those GTA3 and Vice City games sucked, but in comparison to True Crime they looked like some kind of shining bastion of game design. I mean, that first True Crime game was unplayable, I thought GTA 3 and Vice City played bad, and they do, but True Crime made them look good.
As for other sandbox open world games that happen because of GTA3, they're kind of sporadic until after San Andreas. I mean there's The Getaway and Mafia the year after 3, then True Crime after that and Midnight Club (also Rockstar) goes open world. In 2004 you get Spider-Man 2. Then in 2005, the year before Saints Row there's (and this is the best of them, especially the normal shooter ones) Mercenaries: Playground of Destruction, the Godfather and Scareface games, Gun, Destroy All Humans!, and The Incrediable Hulk: Ultimate Destination. There's another True Crime in there somewhere too. I think by that 2005 batch all those with shooting, even if they still had some kind of lock-on system, also had normal third person aiming too like Saints Row would have.
Hearing or seeing the word "grounded" in regards to any Saints Row game always seemed odd to me. The most grounded of the Saints game, the first one, is probably more cartoony than the most cartoony GTA game. It's a series where the biggest side active you do is a bad Burnout knockoff minigame called Insurance Fraud where the goal is to build combos by ragdolling your character into multiple vehicles as they're thrown hundreds of feet by cars. The series at its most grounded is still an over-the-top cartoon parody, and it pushed that more and more with each passing game. The first Saints Row is also probably the most generic GTA copy of everything to follow after GTA3.
Given how the series started with you as basically this Guardians Angels gang, and Volition made The Punisher with Garth Ennis, it's too bad they didn't bring him back to reboot this series since at least before they became the villians you're pretty much meant to be a Punisher gang. Shit, at the least they could've just ripped off their own Punisher game.