You missed a key part of that DMG passage:
"Big monsters typically wield oversized weapons that deal extra dice of damage on a hit. Double the weapon dice if the creature is Large, triple the weapon dice if it’s Huge, and quadruple the weapon dice if it’s Gargantuan. For example, a Huge giant wielding an appropriately sized greataxe deals 3d12 slashing damage (plus its Strength bonus), instead of the normal 1d12."
Large creatures don't automatically add an extra die of damage. (This is why the Trollish Stature feat increases their claw attack to 2d6.) They are capable of wielding weapons that are larger, and therefore do an extra die of damage. Therefore, a large PC would need to also acquire large-size weapons.
I could also see an argument that proficiency-wise, they should be treated as different from their normal-size counterparts as a hand axe is from a battleaxe. So your Large PC would also have to get proficiency with Large weapons, either individually or as a category (sounds like a job for another feat!).