Hello there! I completely understand your concerns about modern meat processing, as it can be quite overwhelming to navigate the grocery store aisles these days.
Between the two options you mentioned, deli mystery meat is generally considered more processed because it often contains added sodium, nitrites, and preservatives to enhance shelf life and flavor. Factory-raised beef does have concerns regarding hormones and antibiotics, but a raw, whole cut of beef like chuck roast is much less chemically altered than processed deli slices.
For a safe and comforting one-pot wonder, I recommend picking up a simple, unprocessed cut of beef from a local butcher or opting for organic grass-fed beef if it fits your budget. Just braise it slowly with some fresh vegetables and broth, and you will have a delicious, wholesome meal for your family without the extra additives.