Tyre failures on aircraft during takeoff are fairly common, and are usually (though not always) caused by flat or underinflated tyres (via the overdeflection and overheating experienced by tyres with abnormally low internal pressures). If an aircraft tyre fails during takeoff, this is dangerous in several different ways:
Flying tyre fragments can damage other parts of the aircraft, as well as, potentially, other nearby aircraft or ground personnel.
A tyre failure(s) can greatly degrade directional control, especially on smaller aircraft that only have a few wheels to begin with.
Run-flat tyres are designed to remain useable even in the event of a total loss of internal pressure (generally by incorporating an internal weight-bearing skeleton inside the tyre’s air chamber); given the potentially catastrophic consequences of a flat tyre failing during takeoff, why don’t aircraft use run-flat tyres?