Correct normal map could be created, to represent the apparent height differences, just like the example height map. Right ?
I feel that the statements in the video from 0:48 to 1:02 are misleading. "They all appear the exact same height, since again there is no height information". "The results from the simple test might make it seem like the height maps are the way to go". These statements make it sound like the normal maps can't convey different height variations. I think this is no true. And for beginners this would be very misleading.
It's true that the results appear so, but I think that's just for this example and also because of the way blender bakes the normals. In fact if we look at the below image, we can see that this normal map would convey 4 different heights without any problem. After all, even the height maps are finally converted to normals to convey the apparent height difference. Also, if we use a software like substance designer and create a normal map from the existing height map, the resulting normal map could convey the height variations just like the height map.