I agree. The majority of DL layers are about feature engineering, not performing classification.
One of the things that interests me about nominal AI applications is the extent to which they're sort of a Mechanical Turk or what I've heard called Artificial Artificial Intelligence. By which I mean it's sold as computer magic, but most of the magic is actually humans sneaking in a lot of human judgement. That can come through humans directly massaging the output or through human-driven selection of results. But I've also been wondering to what extent natural human intelligence is getting put in at the lower layers of the system, like feature engineering.
(That's literally what neural nets with relu as activation unit do.)
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?