It's easy to beat a naive logistic regression model with a good neural network, but the gap often closes once you start trying to tune the logistic model too. (And it's not like the neural networks aren't tuned either--architecture search, data augmentation, etc).
Recent review on medical data: https://www.sciencedirect.com/science/article/abs/pii/S08954...
In many papers, the "baseline" logistic regression model is very stripped down: y~logit(.) but the neural network has had its expressiveness optimized in various ways. People aren't comparing against a 3 layer feedfoward network; there's augmentation and pre-training, architecture search and special learning schemes.
My claim is that if you want to claim that a problem needs the expressivity that (only) a neural network provides, you ought to be devoting a great deal of effort to the logistic regression model too. Make it a steelman, rather than a strawman, if you will.
Imo deep learning is so popular because it "works". For a classification problem, if you try a linear baseline and a deep learning model, and you do a reasonable job of hyperparameter tuning and experimental design, it's likely you will outperform a simpler model. This holds true across many problem spaces.
I think the issue is that modern DL frameworks make it a little too easy to get pretty good performance on new problems. Other techniques generally require more background knowledge to make reasonable modeling assumptions, and still frequently perform worse than a naively applied DL approach.
I think DL will remain, in practice and education, a very popular tool. But it is essential to learn traditional statistical inference and other background to appropriately contextualize DL models so it isn't just some form of black magic.