There is no reason to think that chopping wood is much different than chopping down a wall of ice (AC 12, 30 hit points per 10' section) or a wall of stone (AC 15, 30 hit points per inch) or a Treant (AC 16, 138 hit points). Or any other object or monster with AC and hit points. If you need to have a bunch of exceptions to damage caused by whips, because it doesn't translate to damage elsewhere, then I think there is a problem.
By the way, our 20 dexterity 10 strength guy is exactly as skilled at using a whip as he is with a hand ax (proficiency +2), but somehow his hand-eye coordination, nimbleness and quickness causes a whip to hit much more often and deal way more damage.
I'm all for whips having cool special abilities like grappling and intimidation benefits, but I don't think whip damage is where it should be compared to other weapons or that the rules align with real world usefulness.
I'll agree that it could stand to be a little lower in damage and still be fine; I don't remember much of the original conversation since this thread was from almost 4 months ago.
That being said, it would matter in a game and especially non-combat situations. If a player says he is going to chop down a tree, I'll ask him with what. If nothing but a whip shows up in inventory on his character sheet, then that tree is going to sit there laughing at him. You can't simply equate items to identical usefulness simply because of the same damage type. Arrows and tridents both do piercing, but I'm not going to let a player shoot tridents from a short bow. Can you imagine hanging pictures in your house with a flail, simply because it does the same bludgeoning damage as your carpenters hammer? Good luck.
Combat gets trickier, since a weapon is a weapon, but I would seriously weigh some basic logic in determining if I'd allow a whip to break down a section of stone wall.