Why wouldn't the earth be "always daytime" if the "daily cycle of sunlight" was drawn full-scale? The question is clear enough in my opinion but, I've found out that if you redraw this picture here: 



You'll notice that the Sun is as big as the Earth. But if you scale up the Sun to it's real size compared to the Earth, you would realize that the Earth would covered in daylight ALL THE TIME. So what's going on here?
The sun is really far away. Thus its rays are essentially parallel at the earth's orbit. So, while the diagram you posted is clearly a bit off in terms of the relative size and distance between the sun and the earth, the parallel rays are about right.