It's not just lexical. You can walk — or run, crawl, scurry, roll, etc. — out the gate, out the back, out the exit, etc. You can look — or stare, peer, gaze, squint, etc. — out the window, out the porthole, out the viewport, out the sunroof, etc.
But you can't (standardly) walk out the house, or out the plaza, or out the village, or whatever — all of those need "of". Nor can you (standardly) peer out the box, or stare out the car, or shoot out the bushes — though you can perfectly well shoot at someone out of the bushes, etc.
Apparently out as a transitive preposition has something to do with transiting from within an enclosed space through a limited aperture of some kind.
Mark Liberman at Language Log
Conversely, out of has to do with exiting an enclosure or container.
It seems to me that you walk or come out of X for those X values where you can’t be in X in the sense of inside X or fully within X.” You can be in the bushes, in the house, in the water, in the blue, and in the car, all of which are containers of sorts (or of you), so you walk or come out of those things. You cannot be inside the door or the window, so you come out those things (without of).
commenter Steve Kass, ibid.