The tires are getting hot because of the huge amount of kinetic energy absorbed by the brakes. They also get a bit of heat from being used (friction from landing) but that's not very big: you can design a tire to take that: after all, your car tires don't get that hot from being driven on.
The brake assemblies act as heat sinks for the kinetic energy absorbed from braking: you have the whole brake assembly heating up to hundreds of degrees, even parts at 800-1,000 C. When you are moving, the air tends to stop the heat from going to the tires. And also to keep the brakes cooler by dissipating the heat. When you stop, the air isn't removing the heat, and radiant plus convective heat transfer from the brakes rapidly heat up the tires, causing the air pressure to increase. The plugs blow to prevent a uncontrolled burst. When you are moving there is no convective heat transfer, because of the air speed: as it heats it is replaced by fresh air. So there is only radiative heat transfer.
This is why tire plugs blow only when the plane is stopped, and the brakes are not being heated anymore by braking, but are actually starting to cool. The heat has to go SOMEWHERE after all, and without a constant stream of air, the tires suffer. After all, they are right next to this massive source of heat energy, at high temperatures. Is like being under a grill.
Some brakes have fans to help the cooling when the plane is stopped, but it's not a standard fit.
The landing gear is designed to prevent wheel burst (or fires) for enough time to allow the firemen to get there to either help cool the landing gear, or else extinguish any fires.
Cooling the brakes properly affects their longevity too: ideally you want them to cool slowly enough to even out any hot spots. Water cooling or even powerful fans can be problem because it can cool some parts too quickly: water mist is suggested.
If you look on the Airbus site, they have some very interesting documents for each plane: Aircraft Characteristics Airport and Maintenance Planning.
Airbus Aircraft Characteristics Airport and Maintenance Planning
There's a section that discusses what aircraft rescue and fire fighting should consider. One part discusses how they should deal with brake overheat and landing gear fire: