What can I do about minimizing small shards of ice shooting out the ice dispenser on LG refrigerator?
Looks like its the quality of the produced ice which is easy to break up, try cleaning the ice tray with vinegar then soapy water,  that may reduce the mineral deposits on the tray and reduce surface resistance. 

I don't have a direct answer, only suggestions, since this is quite a complicated question. 

Firstly, if you look at the new ice, fresh from the ice maker, are they solid?, then take one out and see if the ice block is very cloudy, this would make the ice slightly more fragile, since it contains many air bubbles.  you could try turning the freezer to a slightly higher temperature to get the ice to freeze slower. 

If the ice is falling out of the tray already flaky, then the problem is in the tray

I would suggest taking all existing ice out if the unit. cleaning everything then checking if there is a change. 

as a second experiment, you could wait for an empty tray, then manually pour distilled water into the ice tray and see if you get better ice. in that case your water has too many minerals etc.