Here's the bottom line - there's no actual trophy for skipping upgrades in the game. Players have definitely done these no-upgrade runs on their own especially on give me god of war difficulty which is pretty intense, but it's more of a community challenge than something official. The trophy list is focused on normal progression stuff like finishing the story, finding collectibles, and actually upgrading your gear.

Maybe interesting for challenge runners - some players have completed the whole thing using just basic weapons like the leviathan axe and blades of chaos without any improvements. Its definitely doable but you'd need some serious skill especially during those tough boss fights. The game actually wants you to use those upgrade systems and even has trophies like spit shine for powering up your armor, so going without upgrades is purely for bragging rights and personal achievement.