I believe that if system-configuration-features includes X11 then Emacs was compiled with X support.
That value is set in configure by the HAVE_X11 variable being true (look for: emacs_config_features=), which in turn is based on window_system=x11 which you can trace back through a more complicated set of tests.
So it seems the answer was already in the docs on github.
We regard fullscreen as a third layout mode here. An X window in either tiling or floating mode can be made fullscreen explicitly by invoking C-c C-f.
I look for the existence of a window manager using wmctrl -m. If that command outputs anything on stderr I assume there is no window manager and run the EXWM startup.
(when (get-buffer "*window-manager*")
(when (get-buffer "*window-manager-error*")
This is a kind of quirky way of achieving it, but it worked for me.
Having frame-title-format set to its original value:
(setq frame-title-format '(multiple-frames "%b"
("" invocation-name "@" system-name)))
shows this on wmctrl utility to interact with X-Windows manager:
$ wmctrl -l
0x00e00003 0 ES-00002604 /...