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.