The author discusses how linear models are generally more interpretable than deep learning methods, but I'd argue that's actually changing pretty quickly. Especially for large image/sequence inputs (which covers most of the applications that are getting hyped up), linear regressions don't perform very well, and often that performance difference prevents them from picking out important features. Given that fast, scalable methods for feature importance are on the rise (e.g.
https://arxiv.org/abs/1704.02685, which the author mentions), you often get equally interpretable feature scores from deep models that are more accurate than analogous ones from linear models.
Basically, my point is that model interpretation strongly depends on how accurate your model is, and because deep learning models are so much better than linear models for some tasks, it makes sense to use them - even if your primary goal is interpretability.
That said, I do believe that if you ever care at all about interpretation, you should almost never be using multilayer perceptrons (which have recently become part of the widening umbrella term "deep learning"), because they rarely work better than decision tree models or basic linear models (and MLPs are generally less or equally as interpretable when compared to traditional methods).