Regarding the whole debate on nurses, it's important to remember that within the U.S. (and a few other countries, like the UK) the role of nurses has expanded due to a shortage of doctors, with nurses diagnosing some health problems and prescribing medications. This isn't the norm elsewhere in the world, where nursing has remained closer focused on patient care, while physicians remained focused on diagnosing and treating medical conditions.
Quite honestly, we have no idea how the split between physicians and nurses works in the Trekverse, but we shouldn't expect it to hew to either how nursing worked in the mid-20th century or how it works today. I would actually hazard a guess that more of a doctor's work could be replaced by AI diagnosis, meaning doctors would over time take on more nursing responsibilities, blurring the lines between the two.
Chapel is absolutely shown in SNW as being a medical researcher, which isn't something we consider related to nursing. However, she was originally a civilian working on the Stanford Morehouse Epigenetic Project. It doesn't take a large jump to presume she was essentially "drafted" during the Klingon War, and shunted into nursing due to a shortage, despite her original background being a medical researcher.
