how to hide the main menu which is not listed in a block
It seems that you want to lock down specific areas of the site with permissions that are not readily available on the Permissions page. I would address that issue in the following way.

I would update the theme to make the region editable and make it so that the menu was not part of the theme by default.

Next I would install the Context module so that I could display specific blocks, in specific regions, when specific conditions are met.

Finally I would create a Context with the settings:

Conditions


"Sitewide: Always active" 
"User Roles": "Authenticated User"


Actions


"Block": Main Menu