How to identify burnt out component
This is a bite late but I thought I would close this thread. Some of the comments were spot on! I received a new replacement unit a while ago, the picture is of the undamaged component in the new radio.

Knowing now what sort of component I was looking for I went searching on various scrap electronics I had laying around. On an old non-functional AM2 motherboard I found a L1087 component that to my layman eyes seemed like it might be an OK replacement. Since the old radio was broken and I had a new working I figured I could at most make the broke a little more broken.
Soldering the L1087 in place of the burnt 6206A was a success so most likely nothing else was broken and I now have a spare radio if the kids break one.