Thank for the help in finding the issue. Problem: WordPress save the recently opened (or the 1st one) menu as a default one for the current user. If we block such menus and leave another menu only, WordPress shows empty menu instead of one which we allow.
Workaround – deactivate nav menu add-on. Login as restricted user and open allowed menu, then activate nav. menu module back. User will see only allowed menu.
I will research how to fix this and return with update.