An aerogel would be among the worst things to use as a heatsink. They are mostly air, held in place. Air has a very low thermal conductivity, low density, and low heat capacity. Air cooling is used because air is abundant and usually readily available, and relies on convection, which an aerogel prevents.

A much better solution are heat pipes, which utilize convection internally to create very high effective thermal conductivity. Assuming the system is not in continual use, the need for a large radiator can be avoided, by using a tank of water or oil (if electrical conductivity is a concern) to dump heat into. With a tank of water, you can also cool below ambient temperature by using evaporative cooling.