First, yes, that's pretty much what deep neural networks are: layers of shallow machine models stacked atop each other, with each layer learning to recognize patterns at different scale; and we train all layers together end-to-end.

Second, it's not PR! This stacking of layers, when done right, can overcome the "curse of dimensionality." Shallow models like kNN, SVM, GP, etc. cannot overcome it; they perform poorly as the number of input features increases. For example, k-nearest-neighbors will not work with images that have millions of pixels each.

Third, I'm only scratching the surface here. There's a LOT more to deep learning than just stacking shallow models.