An additional thought, water from a free draining sponge could be collected in a water butt or other storage container to be returned by a pump (solar) when it is best required. This would address the issue of excessive roof loading and allow the sponge to provide insulation in the winter and cooling in the summer.
I like the idea, but I see a possible draw back with current design; that without the ablity to drain the sponge or allow air to pass through it, it may become a breeding ground for bacteria, possible even legionaries disease. This would probably worse during damp and cool periods but during the winter the sponge could just freeze into a block ultimately destroying it's structure. You mention that sponges have already been used for this purpose so it may be worth investigating if they were free draining and if not whether my fears materialised.
Whilst your current design may have problems in temperate climates, it could work quite well in the tropics providing cooling by water evaporation. The mats could sit on flat roofs, a common feature of many structures in this region, being feed or sprayed with water.
One other point, as I understand it vulcanised rubber is very difficult if not impossible to reshape or remould, hence it inclusion as granulate in this challenge. And I suspect that none of the other materials on the list can be readily made to act as an adhesive so it may be worth considering non recycled materials for this function.