Did the introduction of cane toads solve the original problem?

Yes their introduction had a purpose. But the toads did not serve it...

They were introduced in Australia to eat small beetles. These beatles were a pest to the sugar cane. But the toads are ground dwellers while The Beatles live high up the plants. So eventually the toads never got to the beatles and searched for food in the surrounding forests and swamps. There they started to reproduce and scince there are no natural enemies the toads are now considered a pest themselves.