The nose will naturally point itself into the wind unless you do something to correct it such as adding rudder and aileron for the flare. So yes a plane will weather vane on its own.
Sigh. As many, many posts here have noted, a plane flying in a steady wind "feels" no wind, and therefore cannot be turned in any direction. Wind is a mass of moving air; a plane flying in that mass of air has no wind blowing "at it" from the side, and will not turn. Gusts, of course, are a different matter, so a plane flying in the wind may turn in any direction because of gusts.
The usual illustration of this is somebody in a hot-air balloon flying an RC model around the balloon. The pilot of course feels no wind. People standing on the ground do. Does the plane? Of course not. Same even without the balloon.
Stick and Rudder, p. 340. "an airplane flying in a steady wind flies no differently from one in still air."