In this case there's no advantage to using logistic regression on an image other than the novelty. Logistic regression is excellent for feature explainability, but you can't explain anything from an image.

Traditional classification algorithms but not deep learning such as Support Vector Machines and Random Forests perform a lot better on MNIST, up to 97% test set accuracy compared to the 88% from logistic regression in this post. Check the Original MNIST benchmarks here: http://fashion-mnist.s3-website.eu-central-1.amazonaws.com/#