Average speed and time are really dependent on the external conditions whereas measuring power is not affected by external conditions.

It may help to think about power in terms of on bike performance. To go faster you need some combination of pushing harder on the pedals or turning the pedals more frequently, which can be measured with power, which is how much energy you put out per unit time. If you training is going effectively then you will be producing more power so it is an effective tool for determining whether or not your training is working or if you need to change things up. BTW this is a very basic description of training with power and does not take into account the many nuances of doing so.