I once damaged my tire by pushing the bike when the tire was flat. It was a quite new tire actually. One day, when I came back to where I had parked my bike, it was flat. Closer inspection revealed that the tire had exploded while the bike had been parked! This was in winter and under a roof, so absolutely not temperature stress.
The point is a damaged tire can explode at any time. You cannot trust them anymore.
Now, it's not a problem if it explodes while being parked. However, if it explodes while you are riding downhill through a corner at 60km/h, it'll get you at least a free ride into the hospital. Unfortunately, tires tend to explode when they are stressed in one way or another, that those situations are those where an exploding tire is most dangerous.
Your life depends on your tires. So ride tires that you can trust.