Well the idea is that even normal wildlife can break out of a shoddy wooden cage. The larger and stronger it gets, the easier it will be to break out of the cage. Building destroyers would get massive bonuses to this as well. Creatures of exceptional size (anything approaching 3m cubed in size) would be completely immune. Even goblins that still had weapons should be able to easily get out of wooden cages. Even copper cages could probably have the bars bent by anything strong enough. So again, the large-ginormous creatures would be safe and the building destroyers would need a legendary blacksmithing working with steel to contain. Only low-threat wildlife would be safely captured by a small fort instead of that early fort building a shoddy wooden cave to trap the resident dragon in before even setting up the dining room properly (done it once). So to capture anything worth while (a grizzly) would require at least a metal industry working with strong metals to contain. For semi-megabeasts, nothing short of masterwork steel with masterwork mechanisms should stand a chance of containing them. Full-blown megabeasts couldn't possibly be contained either.
They'd still be powerful, but would require far more set up time. If you don't have ready access to steel (or even iron), you'd have to make choices to how many traps you want versus equipment for your dorfs. Personally, I don't abuse them. I don't have them up for goblin/zombie attacks and typically just use them to capture wildlife coming out of the caverns. However.... installing a dragon-turret out front is just a little too tempting for me >.> That being said, dragons are supposed to be intelligent creatures- why can we even tame them in the first place? They believe creating wealth yourself is for chumps and the truly powerful just take from the weak. They also get covetous and decide they don't have enough shinies- prompting more raids. It just seems weird to be taming an intelligent creatures like a dog.
Nets. We don't really have nets in DF yet. They'd definitely count as non-lethal combat for a chance to tame critters. Beast-tamers could carry extra food and equip themselves with nets/ropes/chains as they head out into the wild looking for "new friends." Just being able to walk up to an animal and befriend them, though, is magic more than anything else. Sure, we domesticated dogs and cats without long training sessions in cages, but this happened over generations of giving them food, selective breeding, and not attacking them. Bare-minimum, the animal trainer would have to be highly skilled and your civilization very knowledgeable about an animal for the trainer to be able to just walk up to and bro-down with a grizzly. That, or magic. Some type of elven hippie-power to pacify beasts, diety-based spell to calm wild animals, or potion that makes the trainer smell like a friendly beast.