KOLWEZI, Congo—Apple Inc., Volkswagen AG and about 20 other global manufacturers found themselves on the defense when Amnesty International reported two years ago that the cobalt in some of their batteries was dug up by Congolese miners and children under inhumane conditions. Many of the companies said they would audit their suppliers and send teams to Congo to fix the problem.

Their efforts haven’t kept hand-dug cobalt out of the industry supply chain.

At...