From the same wiki article:
The amphora complements the large storage container, the pithos, which makes available capacities between one-half and two and one-half tons. In contrast, the amphora holds under a half-ton, typically less than 100 pounds. The bodies of the two types have similar shapes. Where the pithos may have multiple small loops or lugs for fastening a rope harness, the amphora has two expansive handles joining the shoulder of the body and a long neck. The necks of pithoi are wide for scooping or bucket access. The necks of amphorae are narrow for pouring by a person holding it by the bottom and a handle. Some variants exist. The handles might not be present. The size may require two or three handlers to lift. For the most part, however, an amphora was tableware, or sat close to the table, was intended to be seen, and was finely decorated as such by master painters.
Stoppers of perishable materials, which have rarely survived, were used to seal the contents. Two principal types of amphorae existed: the neck amphora, in which the neck and body meet at a sharp angle; and the one-piece amphora, in which the neck and body form a continuous curve. Neck amphorae were commonly used in the early history of ancient Greece, but were gradually replaced by the one-piece type from around the 7th century BC onward.
Most were produced with a pointed base to allow upright storage by embedding in soft ground, such as sand. The base facilitated transport by ship, where the amphorae were packed upright or on their sides in as many as five staggered layers. If upright, the bases probably were held by some sort of rack, and ropes passed through their handles to prevent shifting or toppling during rough seas. Heather and reeds might be used as packing around the vases. Racks could be used in kitchens and shops. The base also concentrated deposits from liquids with suspended solid particles, such as olive oil and wines.
I'd say by the ancient people's idea the pointed base was a feature, not a bug. The smallish size made it easier to handle than a barrel, and it would always be risky to stack fragile clay vessels in large piles without support in any shape. Ships then were smaller and had less cargo volume so huge stacks of barrel-like containers wouldn't fit.