From a University of Texas study on armadillos and damage they cause.
Most armadillo damage occurs as a result of their rooting in lawns golf courses vegetable gardens and flower beds. Characteristic signs of armadillo activity are shallow holes 1 to 3 inches (2.5 to7.6cm) deep and 3 to 5 inches (7.6 to 12.7cm) wide which are dug in search of food. They also uproot flowers and other ornamental plants. Some damage has been caused by their burrowing under foundations driveways and other structures. Some people complain that armadillos keep them awake at night by rubbing their shells against their houses or other structures.
There is evidence that armadillos may be responsible for the loss of domestic poultry eggs. This loss can be pre vented through proper housing or fencing of nesting birds.
So, the ethics of shooting armadillos might be debatable since most damage could be prevented through other means.
Protecting livestock from coyotes, however, does not seem to have any alternatives except coyote population control, well any means that are remotely financially viable.
Coyote control without killing involves VERY expensive things like producing livestock indoors, herding livestock into indoor pens every night, or building exclusion fences which are VERY expensive.
Guard dogs and guard donkeys (don't laugh, coyotes hate donkeys for some reason) can be somewhat effective but in large open ranges not always reasonable.
Some states are playing with a collar similar in function to a flea collar that livestock would wear, soaked with a chemical compound known to be thought of as nasty by coyotes. The effect of that on the livestock and the beef they produce is still being looked into.
A good number of my family members are still cattle ranchers in West Texas and there just doesn't seem to be an alternative to thinning the population. The M-44 cyanide ejectors were popular for a while, but the chance of a dog getting into one too great, those things are just too indiscriminate. Same with trapping, you never know what you'll get. Live trapping just moves the problem to someone elses land.
So that leaves hunting. If I acknowledge that the coyote population needs control and I acknowledge that devices are too indiscriminate then that leaves shooting.
So, ethically I don't see any problems with coyote hunting in beef country.
As for technique, yes there is a lot of trial and error. Where I grew up the wounded rabbit calls were king, but other types didn't seem to work at all. Took us a while to find exactly the right combination.
That's part of the enjoyment of coyote hunting to me, watching the animals, learning their behavior, basically spying on them to figure out how to beat them. They are pretty crafty animals.