I've NEVER trained for a blown tire in the simulator in almost 30 years of flying. There were always much more serious things to train for.
My company trains for blown tyre both on take off and landing run. The real concern is that crews will abort the take off right up to V1 if the blow-out occurs. The teaching is that after 20 knots below V1 it is safer to continue the take off due loss of braking efficiency. Blown tyres late in the take off run are more common that engine failures and the danger is in the abort as mentioned. To say there are always much more serious things to train for is rather a foolish statement. We train in the simulator for serious non-normals and blown tyres on take off is potentially extremely serious and thus competence at handling this situation has to be certified on the pilots records.