Honestly, the market is so oversaturated with PHDs who are switching to DS I don't see how anyone can transition into it from a different role. I'm speaking for myself as well as someone with a math background, most people don't consider a math degree and 4 years of applying math models as a data engineering experience enough to be a "data scientist". They just weed out anyone without a PHD.

But, what you're describing I would consider "data engineering" (at least how I have been hired to do it). Working through the business problems and pragmatically facilitating data, pipelines, databases, and models to solve those problems. It's less established and less "hot" but, IMO, it's a much more valuable job to most businesses.