Elephant Garlic is more closely related to leeks than to true garlic. Both belonging to the lily family and the genus alium.
The bulbs are divided into large cloves, but there may also be smaller side cloves called corns. These may also be used for propagation. So you can have your cake and eat it.
You may leave the plant unharvested if you like, they do not easily rot and left alone the plant will grow into a bigger clump over time.
You also have the decorative benifit of the flowers for months and their presence discourages pests.
When harvested therey is quite a lot of water in the bulb, so allowing them to try in cool dry conditions may be beneficial.