PhD program in statistics
Certain ingredients are needed to give you the best chance of a successful PhD. One of the important ones is that you and your supervisor have mutual interests.

A second important ingredient, in my opinion, is that you immerse yourself in that environment. It's important to develop a network of colleagues. It helps to spread ideas, start collaborations, get help when needed, and to explore unthought of opportunities.

From what you have said, I think you will be missing out on these two important ingredients if you continue in the same place or if you work remotely.

What is also important is what you to do after the PhD. PhD is required for academic position. But I think you will be in a weak position to get to the next step (fellowships, faculty positions, etc.) if you do what you proposed. In certain industrial positions it can be looked on favourably, not necessarily for the topic you pursued, but because it says something about your personally. Basically that you can get things gone, rise to a challenge, work independently, work as a team, communicate difficult topics and can bring creativity to solving a problems. 

My advice would be to find a machine learning research group and apply for PhDs. If this is not possible why not consider following the topic of your supervisor and keep machine learning as a hobby? You will become and expert in statistic and so you will find manny concepts will translate between the various statistical disciplines. But only do this if you get along with her/him, and you can see yourself studying this topic.

Finally you could try a compromise? Are there applications for "mixing statistics" in machine learning? Can you find one? Is there an unexplored opportunity to do something new?



As a side note I find it ridiculous that PhD supervisor ask the student for topics. This always leads to problems because the student doesn't really have a clue about the research field. There is room for flexibility but often this hides supervisor laziness.