Of the six ability scores, Strength seems to be the one that is hardest to grasp when looking at the inherent size differences of DnD's race/heritage. And to a lesser extent Constitution. Mechanical ways to differentiate size (AC, melee damage, jumping, grappling, carrying capacity, weapon and armor availability) that ignores character race/heritage is one way to tackle the issue. A Gnome could have 20 Strength and gain all modifier and class benefits for having it. But when the Gnome comes up against a 20 Strength Half-Orc they could be at a disadvantage.
I suspect "carrying capacity" can strictly refer to size. A Large character can carry a Medium-size load, without encumbrance, and upto a Large-size load, with encumbrance, and maybe a require a check to do it, if the DM feels the shape of the load seems awkward to carry.
A Medium character carries a Small load.
A Small character carries a Tiny load.
I dont care about encumbrance, but this is a rule of thumb in case there seems to be a narrative issue. It normally only comes if one character is carrying an other character − or carrying items from a hoard of treasure.
Generally, I use size (not weight) during gameplay. I prefer things like Mage Hand and Telekinesis to refer to the size of an item or creature.
Say a Small gnome is carrying a Large horse. If the gnome could lift the horse, it would still be awkward to carry. The gnome seems more likely to drag the horse, which might be painful to the horse.
In 5e, a Large player character is no problem − except for Large weapon damage. Everything else balances well.
For Large weapons, my current thinking is, a Large creature adds a 1d6 size bonus to damage after all calculations. So a Large longsword deals 1d8 plus any class or magic calculations, and afterward adds +1d6 to the total. Huge adds 2d6. Gargantuan adds 3d6. Done. As such, a two-weapon fighter adds 1d6 twice if each weapon hits, yet because this spends a bonus action to do, it might be fine.
Notice, the size of the weapon doesnt matter. Only the size of the creature wielding it matters. The weapon itself is only larger or smaller for the creature to wield it comfortably (thus avoid a disadvantage).