Even if you are only interested in prediction and do not care about inference (like identification of signifcant influence factors), logistic regression can be a reasonable approach: it depends on the amount of training data that you have available.
In the study listed below (sorry, it is in German), logistic regression outperformed neural networks for training set sizes below 20 000 items. Beyond that size, neural networks became better.
Note that there are many application areas where data is not as abundant as in data mining by social network companies. In medicine, e.g., it is not uncommon to have less than 50 test persons. Good luck with training a neural network on that data...
M. Kass: "Textklassifikation mit neuronalen Netzen und klassischen Modellen." Technischer Bericht Nr. 2019-01, Hochschule Niederrhein, Fachbereich Elektrotechnik und Informatik, 2019