I really don't understand why this is so complicated (well I do...it's political) but vets put animals to sleep routinely and very effectively. Why can we just use what my vet uses?
Same here.
The 3 stage lethal injection consists of a painkiller, an anesthesia and then the lethal drug. All to stop any pain, knock him out and stop his heart.
At least get the relevant facts right...
The standard (SCOTUS approved) lethal injection protocol is an
anesthetic, a
paralytic, and a lethal drug.
The paralytic is problematic because it can hide a failure of the anesthetic. Awake+paralyzed+lethal drug taking effect is pretty fucking inhumane.
OK and some other states have been substituting a large does of
sedative since the approved anesthetic is not available. Sedatives aren't anesthetics, though they can work in theory, but practice gets complicated. Remember, there isn't a lot of opportunity for testing and it has to work on the specific condemned person being executed.
As for euthanizing animals...
First off, there are different protocols for different animals because drugs often effect them differently. There isn't a single "what the vet uses" if the vet in question is at all competent.
Second, there has been a lot of opportunity for experimentation and testing of those protocols. Don't forget about all the animals which have been used in research... figuring out ways to humanely and easily euthanize them has been (and continues to be) pretty important.
Finally, with animals it isn't normally a huge problem if the normal procedure fails with a small percentage of them (due to some individuals reacting to the drugs differently). There are alternative procedures available and normally at hand. That isn't acceptable for executions though. and for some unknown reason they never seem to have a plan-B.
PS: A simple summary of the problems with lethal injection as an execution method:
The physiological response to pharmaceuticals is (normally) very complicated.
ETA: If there was a good anesthesiologist with a well stocked kit and no paralytic masking (so the anesthesiologist could do his job correctly), the actual 'lethal' part an execution wouldn't matter at all. As I said before, the problem is that anesthetics involves a lot of skill/art... you can't just follow a recipe and expect it to work reliably.
PPS: The large-animal necropsy at the vet school is very near my lab. If anyone has some specific questions about euthanizing animals, I could drop by and ask them. Just message me.