Why periodic bounday conditions?

In the free electron model (electrons in a box/conductor) why do we use periodic boundary conditions? What the idea/justification for it?

I understand that, to have a unique solution to the SE, we need a boundary condition. We can either choose the wavefunction to be zero at the edges or periodic boundary conditions. The first will give rise to standing waves, the second traveling waves.

I've heard things like:

- Periodic boundary conditions is better, since we have travelling waves which makes the jump to the study of electron transport phenomena easier.

That's pedagogically very cute, but doesn't give me any insight.

- The idea is that the box is very big, so whatever happens a distance L further doesn't affect stuff here and we can simply apply periodic boundary conditions with the physical affects.

Well, the traveling plane waves are infinite in extent and non normalizable. Their position distribution is uniform through the conductor, so that kinda defeats the argument in itself.

And yet, it seems that the application of periodic boundary conditions is very important and leads to certain result wou would otherwise get. Can anyone enlighten me about the wisdom behind this?