Umm, probably not. Yes, diene containing rubbers used in tread compounds (typically NR, SBR, and blends thereof) crosslinked with sulfur can continue "curing" at room temp, albeit at a much, much slower rate. This in theory could result in higher modulus (stiffer) tires. However, some properties, such as tear strength are inversely related to crosslink density (modulus). So I wouldn't assume that a stiffer tread is necessarily more resistant to damage from debris. Keep in mind the manufacturer has likely designed the tire to function optimally as manufactured. Aging only increases the potential for other issues.

I would be more concerned that an older tire had a greater chance of exposure to and attack by ozone and UV. Diene containing rubbers are susceptible to attack even when properly compounded with protective ingredients. This attack results in a less elastic compound. If very severe, surface cracks can begin to appear (this used to be called "dry rot")

Storage in dark, cool, dry areas, away from sources of ozone (electrical motors, for instance) is good advice. If practical, I would store the tires unfolded. Speed of attack by ozone is increased and more severe on strained rubber. If not practical, store the folded tires in a sealed bag. Thats what I do.

CG