Hide protected posts from the default homepage and all archive screens, in the main query, in WordPress.

mattyza-hide-protected-posts.php

add_filter( 'pre_get_posts', 'mattyza_custom_maybe_load_post_hider' );

// Conditionally apply a filter to hide protected posts. The condition is that it must be the main query and either an archive or the default homepage.
// If the conditions aren't met, be sure to remove the filter, in case it is inadvertantly applied.
function mattyza_custom_maybe_load_post_hider ( $q ) {
	if ( $q->is_main_query() && ( $q->is_archive() || $q->is_home() ) ) {
		add_filter( 'posts_where', 'mattyza_custom_hide_protected_posts' );
	} else {
		remove_filter( 'posts_where', 'mattyza_custom_hide_protected_posts' );
	}
	return $q;
} // End mattyza_custom_maybe_load_post_hider()

// Small utility function to modify database queries to hide protected posts.
function mattyza_custom_hide_protected_posts( $where ) {
	global $wpdb;
	return $where .= " AND {$wpdb->posts}.post_password = '' ";
} // End mattyza_custom_hide_protected_posts()