If the goal is to "toss a heavy object to hurt an opponent", that seems like a pretty clear application of the Shoot skill to me. It governs all the ranged weapons.
I wouldn't really use Athletics for it, at any rate. If you want to emphasise "these things are heavy" then it would be better to use Physique, as that represents muscle and general toughness.
(Although having a good shoot skill usually implies the character already has a good ranged weapon to make these attacks with, so throwing Clay Jars is probably more something for the less able ranged attackers to do.)
As for throwing into melee; that's allowed and works the same, although it might be reasonable for your opponents to Invoke being in melee in order to up its defenses. (Or for your ally to Invoke it to help you by shoving the guy in front of the jar)
That said; throwing something really big and heavy is unlikely to inflict actual damage. It seems far more fitting to use them to Create Advantages (depending on the contents of said barrels) as enemies jump out of the way, clouds of flour float around the room, grain rolls over the floor or swarms of angry scorpions sting everyone in range. (Legend of Zelda features jars that are full of scorpions for no reason, right?)