"At what point did the breeding of animals and manufacture of bronze become "scientific?" Was it as soon as any notion of God or intelligent design was left out of the picture?"
Well, now. Animal breeding is pretty simple, really. No science is really required. You merely put male and female animals together. When the females are in estrus, the male will pretty much take care of it for himself.
As for bronze, it wasn't invented by the Jews, you see. The folks who invented it had some other diety or three.
Bronze is interesting, metalurgically, since copper and zinc often occur in the same mineral deposits. The first bronze probably oozed out of an oven used for baking bread when the baker accidentally put too much wood in the fire and got it too hot. Ruined the bread, I suppose, but that little trickle of bronze would have been recognized as a useful metal.
Before the discovery of bronze, humans used native copper, as found in nature, to make tools. Silver and gold also occur as native elements in nature, so they were commonly used for ornaments, or for lovely things like the Ark of the Covenant.
Isn't that interesting? No intelligent design or creationism required. No science, either, although modern science grew, in part, from the experience of those working with metals and with the breeding of domestic animals.
It's an amazing world.
So we should assume that at no point did it become necessary for humans to research and hypothesize regarding ways to breed animals for a living or make use of metals; that these practices do not involve, and never have involved, science.