Aluminum is more prone to stress-wear than steel is. The tires would literally come apart as the aluminum cords began breaking due to heat and flexing stresses. This is exactly why you don't see come-alongs (hand winches) with aluminum cables in them.
It's a novel idea, but something resembling carbon-string (carbon nanotubes) would be more likely put to use. (read:
space elevator)
Also, when you consider the weight of the rubber in the tire, vs. the "other" in the tires, you've considered more than 50% of the overall weight, in a single component, making lighter rubber compounds a stronger concern than which metal is used for the 2 or so lbs of banding in the tire.
Most tires have steel, nylon and/or rayon, polyester and/or fiberglass, and rubber in them.
When you look at your sidewall, it will say something like "tread plies 2 steel + 1 nylon" "sidewall 1 fiberglass + 1 polyester" making a total of 5 components that comprise less than 50% of the tire's weight. The heaviest reinforced part of the tire is the bead, and even a cutaway of the average passenger tire bead only weighs several ounces with the rubber removed.