Why does a cold saturated sugar solution melt chocolate?

If there is no heat, the chocolate is not melting; something else is going on. The closest thing I could find to an explanation was on this site about how chocolate is made:

Cacao is a rich source of insoluble but water absorbent fibers (cellulose, primarily) which tends to cause the particles to draw in the water and swell up, further disrupting the texture. The cacao particles then tend to push away the cocoa butter.

Of course, this is in reference to pure cacao, an ingredient in chocolate making, rather than to a finished product like that egg in the video. Still, it points to a mechanism by which the liquid in the sugar solution might interact with the chocolate in the egg that is neither melting nor dissolving the chocolate, but might make it look like the egg was melting or being dissolved.

I also found some discussion of sugar bloom on chocolate that's exposed to most conditions. The problem there is that the moisture is dissolving the sugar out of the chocolate; in your scenario, the sugar solution is already saturated and the change is taking place very rapidly. I would tend to give more weight to the explanation about the cellulose in the cocoa solids.