I feel like modern ML is finding out that you can apply the chain rule to calculate derivatives cost effectively and then scaling up a really really really big equation to then iteratively converge on the answer you want.

It's an experimental discipline implemented _using_ computers. Programming is necessary but definitely not sufficient.

I'm biased, but in my experience, stats, the math-heavy end of economics, and computational physics/chemistry/biology are all roughly as good preparation as a traditional computer science course. (Having research experience is particularly useful.)

That’s mostly the case — the line between ML and statistics is blurry. However the concerns of a practitioner of machine learning and a practitioner of statistics can be different. The ML community has a greater interest in stuff like “can I come up with a procedure to learn this thing.” Lots of practitioners don’t care about why something works as long as it works (a theorist can figure it out later).

A statistician might be more concerned with getting the modeling assumptions right and applying methods in a more rigorous manner. They might not push the envelope as much in terms of application, but that’s because they have a higher standard they hold themselves to.

However, this line is incredibly blurry and there’s plenty of more detailed write ups by people who have spent time in both communities (I have a bit of bias in that my exposure is primarily to the ML community). There are people that do applied work and theoretical work in both communities and there are good and bad practitioners in both communities.