How to hide Newsletter plugin submenus from the Dashboard?

Use the action admin_menu as opposed to admin_init. You also need to adjust your use of remove_submenu_page to match the way Newsletter adds menu items:

remove_submenu_page( 'newsletter_main_index', 'newsletter_main_index' );
remove_submenu_page( 'newsletter_main_index', 'newsletter_main_main' );
remove_submenu_page( 'newsletter_main_index', 'newsletter_main_diagnostic' );
remove_submenu_page( 'newsletter_main_index', 'newsletter_subscription_profile' );

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