The reason that tranquilizer darts were not used at the time this movie was made was that in the early 1960's they were not available as of yet. They are a fairly modern invention. They are not also entirely safe. The daarts themselves can cause permenant if not fatal harm. Also, the drugs used can also cause serious injury and or death. Also if an animal is only grazed by the dart then there is the chance that it could escape a capture team and die or be injured in the drugged and drunken state. Many animals do not tolerate tranquilzers well at all.
Here is an idea.. trans that are considered safe for horses.. ace.. can cause male horses to have serious life threatening injuries to their privates.. to remain out, rather than retracted, it is a about a 1 in 10,000 injury but it happens, also that particular trank known as ace, has been known probably 1 out of 10 times to make horses go the opposite way, making them really hopped up. Another trank used in horses called Rompun, often makes horses what is known as jkicky, they will kick anything they sense move behind them.. see where I am going with this, these are drugs that have been used on hundreds of thousands if not millions of equines and they can have serious side effects or problems, think about using darts on animals that they have not been extensively tested or used on, in the wilds.. which is why darting is a problem. Even now, most of the time, problem wildlife are trapped or killed rather than darted... or at least in the US.
I love the movie and I am an animal conservationist, but I also realize when the movie was made and accept it for what it is. Also in all honesty for many people other than the old wild kingdom you dont see the african plains looking like that anymore, it is a step back time. game is not nearly as plentiful now..