I'm wondering if that is possible, so that I can use some X11 window manager for everything.
No. Native OS X apps don't call down to X11 protocol and no one (Apple or other) has implemented any sort of shim/translation layer/conversion tool to port over the API.
It would surely involve slowdown, reduced acceleration, possibly loss of fidelity and loss of functionality.
You could have an alternate login to X11, but use a non- Apple X11 VNC client to call Apple screen sharing.