Hi!

Would like to have a completely private blog. Use the plugin to password-protect the Homepage. However. when I load it, I see not only the password box but also the menu with all the pages. Strange. 

Ok. I change the visibility of pages in the menu from "public" to "private". No menu now, which is good. But when I enter the password for the Home page or login as Administrator. I no longer see these pages in the menu. Which is even stranger. Doesn't "private" mean that only I should see them?

Will appreciate any advice about these settings.

The Password Protected Pages plugin is made to solely prevent access to the content of single pages. It will neither protect common website parts like the navigation or the footer, nor other pages.

Typesetter doesn't have built-in tools to restrict access to a whole website.

If you want your site to be entirely private, directory protection using HTTP Auth (e.g. via .htaccess + .htpasswd with Apache), is an easy way.
Most webhost control panels have tools to manage them.