Restricted access to the WordPress page but still getting errors: 'Sorry, due to an error, we are unable to fulfill your request at the moment.'
I have found a partial solution...

To wrap the CiviCRM shortcode in a Members shortcode (which would have been the 'old' / pre-gutenberg block way to do it)

[members_logged_in][civicrm component="user-dashboard"][/members_logged_in]

The content is then restricted to logged in users (in this instance) and no visible errors are thrown, though while the form works it appears some Javascript/CSS is not loaded on the page when the shortcode is nested - which others have also come across here: Request login on edit mode profile form.