I would recommend doing a CS PhD and take statistics courses, rather than doing a statistics PhD.

For examples of promising research areas, I recommend taking a look at the work of FLI grantees. I'm personally working on the interpretability of neural nets, which seems important if they become a component of advanced AI. There's not that much overlap between MIRI's work and mainstream CS, so I'd recommend a more broad focus.

Research experience is always helpful, though it's harder to get if you are working full time in industry. If your company has any machine learning research projects, you could try to get involved in those. Taking machine learning / stats courses and doing well in them is also helpful for admission. Math GRE subject test probably helps (not sure how much) if you have a really good score.