Per,
What is the source of your thesis?
I am afraid this it's not correct. A sine wave can be well played back as you say. Every short time signal, transient, even the shortest impuls can be seen as a sum of many long term sine waves at different frequencies (Fourier). Thus they can be perfectly reconstructed and played back as accurate, if all frequency components (that is the sine waves) are below half the sample rate.
The video does not state the opposite.
There is no doubt either in the comment posts.
However, the video is somewhat missleading as it shows the digital signal in steps.
But in reality digital signals are not stepped when they are converted to analog.
Feel free to ask me any question about this whole sampling topic.