Switty uses a standard WordPress comments implementation, so the following, added to your proper child theme functions.php, will erase comment author names.
The function could be written as an anonymous function – so very short – but I figured you just might want to return something in place of the name, plus anonymous functions are kind of un-WordPress.
/**
* OMIT COMMENT AUTHOR NAME
* In standard WordPress Comments
**/
add_filter( 'get_comment_author', 'wpse_get_rid_of_comment_author' );
function wpse_get_rid_of_comment_author( $name ) {
return '';
}
However, it will leave an orphaned “says” behind, which might look fine coming after your avatars… or I guess might not! If you don’t like it, you could eliminate it with CSS ( .says {display: none; ). Otherwise, things might get a little more complicated, though you can find some ideas here: Remove “says” from comments