I've only heard of massaging kale. If you rub the greens together they get softer, darker and more tender and useable in a salad. Never done it with anything else. Wilted is steamed or blanched greens. Wilting definitely changes the texture and can make the greens more palatable and less bitter. I won't eat raw collards. I also always salt and squeeze my cooked spinach before using as it knocks out the bitterness.
Assuming you're talking about ggaennip, also known as shiso, it's quite common to use it raw or salt-pickled. There are two common varieties, red and green-leafed, and to my taste there are some subtle differences between the Korean, Japanese, and maybe Vietnamese varieties.
In Korea and Vietnam, ggaennip is frequently used to wrap foods (ssam bap, for example, or the Vietnamese turmeric crepe). Shiso can be used to wrap foods, as a garnish, or chopped and served on some things, like grilled chicken (yakitori) as a flavor contrast.
I don't see much point in cooking shiso or yukari, but I've cooked the red pepper and salt pickled version sold in cans (or sometimes in the freshly made kimchi section) by adapting it into a sort of variation on the mediterranean dolma. It retains much of the flavor, but is different than the raw leaf and mellower than just the pickle it starts from.
I do appreciate shiso tempura, so I'm not dogmatic about not-cooking, and there is certainly precedent for some cooked applications. I've also made an agedashi-doufu, deep fried tofu in soup stock, which had shiso added before it was coated for deep frying.
Shiso/gaennip is now popularly used in a drink that has added lemon juice or citric acid, but I've only seen this as a concentrate or prepared beverage, so it's possible that it requires some industrial techniques to retain the flavor when making a syrup.
Dried salted shiso, called yukari in Japanese, is also used raw, but often mixed into warm rice either as a topping or to make onigiri.
Best Answer
The physical massaging of the greens starts to break down the cell walls (made of mainly cellulose in plants), rendering the greens more tender. The addition of salt could be to help draw our more moisture from the greens, or it could just be for seasoning like vinegar or lemon juice.