Logistic won't do anything useful for text, to be sure, although an HMM often will (or if you have continuous-valued sequences, a Kalman filter often will do the same). Logistic or multinomial can be tremendously handy for picking up interactions between measurements that can be followed up on and/or expanded in the limited-data case.
I think that the nonlinearity is what really sets apart problems better handled by NNs (not just nonlinearity, but nonlinearity that resists any sort of linearizing transformation), even for lowish-dimensional data. If you look at a linear fit plus a relU, you're just tacking a hinge loss onto a linear/logistic fit. Stack a bunch of these on top of each other and you have a universal function approximator, for which the goodness of fit is limited by the data. If you don't have a crapton of data, the fit isn't likely to be a lot better than linear or transformed linear. If you do have a crapton of data with nonlinear relationships, the implicit structure can be better captured by the flexibility of an NN. But of course you can also spend a lot of time training and debugging the fit, when it might be possible to quickly fit and diagnose a low-dimensional linear or additive model and put it into production. For a long, long time, the most popular "machine learning" method in the valley was logistic regression :-)
For image classification the way people expect it to be done, you are absolutely right (CNNs are incredibly good at this when given enough labeled data). E.g. for relating histological images to genetics or other markers, there's almost no point in not using a CNN with or without a denoising autoencoder in front of it. For low-detail or sparse-and-low-rank mixtures, often you can use compressive approaches to get a lot faster training. But I'll not argue against CNNs for the general image recognition case.
Linear or logistic is typically a great start, and as you note, it's very general. If after trying the simplest thing that can possibly work (linear or logistic), you need better performance, or the linear models can't give you useful answers, ratcheting up the complexity is a reasonable response. You do need a good deal of data to make the latter step worthwhile in most cases. I see a lot of people skipping the first step or ignoring the need for lots of data, and these are the people who get in trouble.