Based on what I know, probably Computer Science is more suitable for both "research in machine learning ... developing strategies for automated stock trading" and "engineering with some pure mathematics (background)" than statistics. If you are aiming for a job in private sectors, they would evaluate you more on "what you can do" rather than "what is your major". And if you search positions posted on glassdoor or linkedin, actually employers don't care whether you are stat PhD or cs PhD or ECE PhD (and I know a couple of ECE and physics PhDs working in Wall Street or HF in Boston now).
Another practical issue is that most professors in statistics departments are not focusing on financial statistics (which is why I had a hard time searching for potential supervisors). And as far as I know, among these professors, not many of them are implementing machine learning techniques, and vice versa.
