Deep learning: input -> feature extraction (auto) -> model selection (auto) -> output
So take a DL image classifier. The convolution + pooling layers perform automatic feature extraction. Back to OPs point, why use something like DL when you've already engineered your features?
Statistical modeling: input -> feature extraction (manual) -> model selection (manual) -> output
Machine learning: input -> feature extraction (manual) -> model selection (auto) -> output
Deep learning: input -> feature extraction (auto) -> model selection (auto) -> output
So take a DL image classifier. The convolution + pooling layers perform automatic feature extraction. Back to OPs point, why use something like DL when you've already engineered your features?