Why is the voltage wave magnitude constant along a transmission line that is perfectly matched to a load?
You understand that for a standing wave in a T-line, the amplitude depends where you are on that T-line.

Now the wave is not standing but travels. So pick a point on the T-line and you see the waves passing by. They all have the same maximum amplitude, right ?

I could move to the left or right on the T-line but I would still see the same as the waves are all the same and they keep passing by.

Why would there be any amplitude peaks ? For that to happen the wave and its amplitude has to be influenced by something. For the standing wave it is the reflected wave combining with the original wave resulting in a combined wave which appears to stand still.

Note that a wave cannot stand still, it is just the vector-sum of two (or more) waves which appear like the wave stands still.

A T-line which is perfectly matched looks like an infinitely long T-line to a wave that is travelling in it. The wave experiences no outside influence/interference because there is none. So it will have and continues to have a constant amplitude.

At the end of the T-line the wave's energy is dissipated in the load (assuming a resistive load), the wave is no more, it is converted into heat. So it cannot reflect and it will not interfere with the other waves.