The problem doesn't have anything to do with GSX so, clearly, a GSX reinstall won't fix anything.
What you refer as the "GSX menu", is in fact the standard Simconnect menu, shared by all application that have a menu, and they don't have any control over it (other than listing which entries it should have) so, to confirm the problem doesn't have anything to do with GSX, you can try another app that use the same menu, for example SODE. Press TAB+S and see if the SODE menu appears.
The usual reason why the menu has problem, is updating P3D using the Client only. Try to update using either the Full installer, or the Content installer.