The term is “chemical immobilization,” and the tool for this procedure is a dart gun that shoots a syringe-like dart that carries some quantity of a drug. How much? Well it depends on what the target animal is--the darts start out empty, and some poor vet tech does math based on what drug the vet says to use (different drugs are differently effective on different animals). There is no phaser that you can set on “Stun.” You can’t knock an animal out without the possibility of accidentally killing the it with the dart..
2. Gotta get the dose right.
An overdose or an insufficient dose is not an uncommon outcome. Especially since the animals in these situations tend to have ELEVATED ADRENALINE LEVELS which makes delivering a precise drug dose very tricky.
3. Gotta be a pretty good shot to do it right.
The person firing the dart has to hit the target: usually the ass of the animal. What if you miss the ass? You could hit a bone and break it, puncture a vital organ or put out an eye. Sometimes the dart hits the target and the stopper doesn’t slip, so the drug stays in the dart. This results in a slightly more terrified and pissed off animal.
4. Once you hit the target, the animal falls gently to sleep in a few seconds, right?
No, the animal might jump from the pain (these are big fucking darts), or turn and bite, or do whatever else an animal in sudden pain might do. If the situation is a person in an animal enclosure with a dangerous animal, pissing the animal off is not necessarily the right choice. Depending on--well, everything: the drug, the dose, the location of the shot, the kind of animal, the size of the animal, the animal’s agitation level, etc etc--depending on these, it may take several minutes for the animal to lose consciousness. In the meanwhile, you may have an enraged, drunken gorilla staggering about.
5. You gotta protect the people.
There are 7 billion humans crawling around on the planet like aphids on a doomed plant, and only about 3500 tigers left in the wild. But if a human falls into the tiger exhibit--even a shitty drunk asshole of a human--you better believe there are no plans to “tranquilize” anything. The tiger count goes down by a significant percentage, and another dumb human survives to fall into something else another day. The job is always to protect human life, no matter how precious and rare the animal life is.