Q: Does an aisle runner serve any actual purpose? Our ceremony site is fully carpeted, so I'm not sure whether this is something we actually need.

A: An aisle runner is definitely a want-to-have, and not a must-have. Some brides and grooms simply like the look of them -- for example, if your ceremony site is carpeted in a distinctly non neutral color (say purple, and your colors are red and silver) and you want to cover it up.
Other couples like to use runners as another way of personalizing their day -- monogrammed aisle runners continue to be trendy, and can often be repurposed into something else (for example, a fancy tablecloth) post wedding.
If you are marrying outdoors or in an unconventional space, the aisle runner could be extremely useful, both to demarcate an aisle but also, in the case of outdoor weddings, to keep your gown clean. Because you're inside, it's your call.
Another option: If your site allows it, you can always opt out of an aisle runner and instead go for flower petals, either carpeting the entire aisle, or running along the edges.