How old were you in the early 2000s? If you see him out of context, you may not pick up on it, but in context, there were a lot of characters like that in the 90s. Those poor tortured souls. Shadow the Hedgehog comes to mind. Even characters that were previously nice and friendly, the Prince from Prince of Persia or Jak from Jak and Daxter, got an attitude adjustment to fit the mold.
I was 25 as of May 2000, and was a comic book reader until 98'. I'm certainly aware of the archetype you're referring to. I remember the ad's for comic stores within the comic books themselves, and one store in particular kept using the phrase "grim and gritty for the 90's" in it's description of multiple books. It was so overused, it briefly became a "thing" my friends and I would say among one ourselves. As for Kratos, I just never picked up on that vibe. To me, and I know this is corny, it felt timeless. Not dated.
I put 90s by accident actually. It kind of started in the 90's with the over the top violence gigantic muscles and anger and giant weapons (Spawn/Wolverine/Punisher/Venom/Cable if you were into comics). The 00's was kind of an extension of that trope. They still had the dark past (Kratos' origin is pretty similar to the Punisher, and even more similar to Old Man Logan, although that was a bit later), but at least they didn't look visually as silly and over the top. But darker heroes were very much a thing.
I honestly didn't get very far into God of War (I think only to the part with the fortress where you had to move the whole thing?) but it seemed like Kratos' solution to feeling bad about being a prick was to be more of a prick. Gameplay was pretty good, but I have a hard time finishing a game if I'm not at least somewhat invested in the characters. I felt the same way with the DMC reboot where the gameplay was very solid, but the lead was so annoying I just gave up. I've been curious to try the other games, but if he's like that throughout the series, maybe it's not for me.