The reason is quite simple. Plastics have a lower specific strength (strength to weight ratio) than metal alloys, which in turn have a lower specific strength than various carbon composites. There are some other reasons too, such as resistance to temperature extremes, a spaceship is exposed both to high temperatures and extreme cold and metals will generally deal with this better than plastics. But just the strength to weight ratio is reason enough to prefer carbon composites or metal over plastic.
Moving onto airloy/aerogels. They are being used or developed by Nasa, as in here or here, including for use as inflatable heat shields. The extremely high insulative value of aerogels would make them ideal for such a purpose, if made out of an appropriate substance.
Also some plastics already have very high applicability for use in space. One example is Kevlar which has many excellent properties, Kevlar has a specific tensile strength 8x greater than steel and is moderately resistant to both heat and cold. Because of it's great tensile strength Kevlar is often used in whipple shields. Kevlar and other high-strength plastics are ideal materials to be used for future inflatable space habitats. Kevlar would also be ideal for tethers. The weakness of kevlar is very low compressive strength and it is moderately vulnerable to degradation from UV radiation. Plastics are also more permeable than metal due to the long molecular chains. Composite materials can exploit the best properties of different materials, and most composites have a plastic component.