Rubber tyres are not made out of what we normally think of as rubber, ie, tree sap. In fact rubber is in almost every case plastic. Plastics are hard, even the soft ones, to make them soft you add a chemical called a plasticiser. For instance, freezer bags are made from plasticised PVC, whereas window frames are made from unplasticised PVC (hence U-PVC).
The softer you want the material the more plasticiser you add, to get really soft materials for gloves etc.. you might add 50% by weight, for tyres around 30%. This plasticiser does not bond with the molecules of plastic but instead holds them apart enough so as they can move relative to each other and hence be pliable. Given time the plasticiser leaks out (migration and leaching), leaving behind a hard plastic that cracks like you would imagine it would. To avoid this, plasticised materials need to be masticated regularly to redistribute the plasticiser around the plastic matrix, hence why car tyres on a static car perish and those on a used car do not, this is what has happened in your case probably, if you are not going to move the saw much, go for solid tyres.
Aidan