Debatable... I've heard there were a lot of failures because the government couldn't get a hold of a quality drug because none of the drug companies wanted their drugs to be used for killing.
Allow me to correct: Lethal injection, when done right, is swift and painless.
There is also potential for failure with firing squads, the largest issue being that the initial shots do not fully kill the victim - either due to poor execution (no pun intended) by the firing squad or inadequate firepower. Either way, there is potential for failure with both methods, but lethal injection is obviously much less gruesome and therefore appears much more acceptable in the public eye.
we should have something that ruptures the brain stem
quick and possibly done without a hole in the face
That's called a shot to the back of the head, and bares the same issues as a firing squad. Injecting an explosive pellet into the victim would also be pretty disgusting when it broke the thin skin in the back of the head, and would as well be impractically expensive.