We have the same issue with the newest SignagePlayer version 4.34.35 and airVersion 3.6.0.6090 on Windows PosReady 7 (based on Windows 7).
Since days i am trying to get it working (in kiosk mode it has to run only one video then it has to go back to an swf-animation). With internet connection all is working.
I've also tried to solve it with the offline updater. After restart and with no internet connection, the player shows only the player starting window with title "SignagePlayer" and white content (no start, stop or import button).
Do you have updates on this issue? The customer is very upset, because we do not manage to solve such a simple task. By the way, we are an enterprise user.