I hear you on this, been fighting him all day and not one gem. I wish they would implement a trade system for these rare materials, at least a gem for gem trade. It's annoying when I fight Zinogre 2 times and get 3 Jaspers, but I fight Brachydios 20+ times and not one gem.
Anyhow, if they follow the same rule as plates, its 3% to get one from rewards, 3% from carving tail, 2% to get one from breaking head (?), another 3% chance from capture rewards, and finally 1% from shiny drops (slime puddles sometimes leave them).
One of my gems did come from break rewards on a kill, so im assuming the head. Also i did get one from a slime pile.
Apparently OP for his entire argument does in fact put his fingers in his ears and go "LALALALA" the entire thread. Fascinating.