In particular, I think the dichotomy stems not from statisticians neglecting what works, or having a narrow mindset, or whatever.
It seems to me that it stems from different goals:
* business seeks to predict and classify
* science seeks to test hypotheses
And statisticians used to focus on the latter, for which you need classical statistics ("data modeling", or "generative modeling", as Donoho calls it), don't you?
And for prediction and classification, sure, there are the classical techniques (regression, time series (ARCH, GARCH, ...), Fisher's linear discriminant), there are Bayesian methods, newer statistical stuff such as SVM, and ML techniques such as random forests.
However, it's just driven by different objectives. As the commenters state, Efron: 'Prediction is certainly an interesting subject but Leo [Breiman]’s paper overstates both its role and our profession’s lack of interest in it.', or Cox: 'Professor Breiman takes data as his starting point. I would prefer to start with an issue, a question or a scientific hypothesis [...]', or Parzan: 'The two goals in analyzing data which Leo calls prediction and information I prefer to describe as “management” and “science.” Management seeks profit, practical answers (predictions) useful for decision making in the short run. Science seeks truth, fundamental knowledge about nature which provides understanding and control in the long run.'.
So, different objectives call for different methods. And, certainly 20 years ago, statisticians were mostly focusing on one rather than the other. Ok. So?