Not a direct answer to your question, but a personal experience. I got a PhD in EE and am now one semester away from finishing my masters in CS. Three years after my PhD, I changed careers from EE research to a data science. My EE PhD provided me background in linear algebra, partial differential equations and numerical computations. The primary motivation for me going into MSCS was to develop intuition in certain areas which I wasn't exposed to, but became important in my new career: combinatorics, statistics, probabilistic modeling, optimization, operating/distributed systems, and computer networks to name a few. Following a masters curriculum guided me on what areas are (roughly) considered important and also brought a (forced) discipline to my studies.
If an application of MS followed by a PhD crossed my desk, I'd laud your discipline, but also probe on why you went that route. You need to have a good answer for that.