What you're talking about is a skip-reentry, where you cut a series of passes through the high atmosphere to bleed off speed before making the final descent. But note that there's no need for wings, and not really a U-turn happening here -- from the perspective of an outside observer floating in space above the north pole, your orbit is just a curved path through the atmosphere that curves less than the surface of the earth does. If you watch your altimeter as you do this, you will appear to descend and then rise back into the sky, but that's just because your altitude is being measured relative to a ball instead of a flat surface. The vessel doesn't have to do anything to make this happen, it's just their orbital path, which happens to pass through some air along the way.
To explain that point more, consider an eccentric orbit that doesn't even touch the atmosphere -- at one end of the orbit it's 30,000 miles above the surface of Earth, and at the other end it's 80,000 miles away. Is the spacecraft "making a U-turn" on each orbit? Not at all, it's just going around in an ellipse. But if you watch the radio altimeter, it says you're going up and down all the time.
Anyway, there are several reasons we don't use skip-reentry paths. Skip-reentry is a trade-off. You can reduce the immediate heating but you have to stretch it out into a low-and-slow bake, which is actually harder to deal with for a spacecraft. Contrary to popular belief, space is not cold (at least, not the way we would think of the word), and things in space (or the extremely high atmosphere) cool very slowly, so if you give the hot shield time to cool, the easiest place for heat to go is into the cabin, and you're going to have to plan for how to manage that. You really want your reentry to be as fast as you can make it without causing damage to the ship or crew. It's usually better to just build a more robust heat shield that can power through the worst of it and get down into the thick air that will convect the heat away, instead of a lighter shield that requires you to tippy-toe into the atmosphere.
There's also a safety issue with a long reentry path. It requires your vessel to stay stable and operational for longer in between between "safely in orbit" and "safely on the ground". You need more air, more battery capacity (since you had to ditch or stow any solar cells before reentry), and potentially more fuel for the control jets. That makes it more risky in general -- there's more time in the critical zone for something to fail -- and it makes the vessel less useful in emergency situations. If something has gone wrong on orbit and you need to get down as soon as possible, a lazy multi-hour reentry path that swoops through the atmosphere multiple times is probably a lot less desirable than just pushing through and getting to a place where you can get assistance.