I suppose you could say statistics is less "empirical" than ML in the sense that it is axiom-based, whether that is a normality assumption of predictions about a regression line or stock prices following a Wiener process. By contrast, ML is less rationalist by simply reflecting data.

I don't think that's true (or maybe I misunderstood?), I guess your comment "simply reflecting data" means fitting data with a very flexible function (curve)? There are very flexible distributions to fit almost any kind of data e.g https://en.wikipedia.org/wiki/Gamma_distribution or with a composition of them, but as a practitioner you still need to interpret the model and check if it does represent the underlying process well. Both statistical inference and ML are getting there using different methods.