It's very simple to figure out the intent of the rule, I think. The target of a grapple can be no more than one size category above you.
A 20 Strength Human Fighter can carry 300 pounds.
A Large-sized creature is at minimum 10' tall. Extrapolating from human average height/weight and the square cube law, that's going to be 896 pounds.
No medium-sized character could carry that weight, thus if that were a requirement, no such character* could move a large-sized creature.
*Possibly with Powerful Build, but no race had this trait when the PHB grapple rules were written.
If the intent was for players to not be able to move large-sized creatures in the grapple, the grapple rules would say so, not force you to go to another part of the PHB to determine an unstated restriction.