I have reinstalled iTunes and tested it in another pristine account as well as in my regular account. The iTunes Visualizer (not the Classic one) will work in a window (not full-screen) in both places. When I choose full screen (ctrl-cmd-F) the Visualizer window grows to full screen but goes black. When I press the key combo again (ctrl-cmd-F) the window shrinks but stays black. If I close that window, I can start the visualizer again in a new window, but cannot make it go full screen. I can stretch the window to near full screen, but the title bar and the screen menubar are still present.
Perhaps this small window functionality was present before the reinstall, but I hadn't tried these moves earlier, just the fullscreen.
Do you have any idea what could be limiting the fullscreen rendering of the Visualizer? My screen is 1680x1050 and I've tried all the smaller sizes that are available with no change to behaviour.