I mean, there's a big difference between highly processed stuff like bologna, vs. stuff like ham that is basically just deboned, cured, cooked, and sealed up in a single piece to be sliced at the deli.
If you stick to the cold cuts that are just hunks of whole meat in a recognizable "animal part" shape -- ham, turkey, roast beef, etc. -- those seem perfectly fine. Heck, some of them (like roast beef) your grocery store might actually make in-house.
I also find the dried cured meats to be totally fine -- salami, pepperoni, etc.
When I see ham that comes in a rectangle, or stuff like bologna or liverwurst, then I can understand being a bit more suspect...