Never understood the mystery behind folding shirts. Then again have been folding laundry for more years than I care to remember.
Large professional laundries have special machines to fold laundry, even shirts, so trying to imitate their results will take lots of practice, and even then. Also commercial laundry press shirts flat, usually with lots of starch which makes folding easier as well.
Bottom line is no matter how well shirts are folded, they are going to wrinkle around sleeves, and perhaps lower torso. Cheap laundries know this and will often do a sub-par job ironing on shirts that are ordered to be boxed and folded. This way they can blame the wrinkles on shirts being folded.
Commercial laundry standards for boxed shirts is to have at least the 8x14 or so top/chest parts looking perfect, as this is the area which shows when a man wears a jacket. Wrinkled sleeves are covered by said jacket, and the lower bits are tucked into a man's trousers.
Do have one of those "flip and fold" thingies, purchased cheaply and used once. It was more effort than it was worth so sits unused and stashed away.
L.