Display tags with random thumbnail from selection of posts with that tag

I am thinking about leveraging the wp_cron and transients.

You can create a helper function which will do the heavy lifting and do all the queries you need, but schedule it to be done like every 12 hours – twice daily (or more, depending on the frequency of new posts) using wp_cron API.

Basically, this is what you need:

Helper function and WP cron job

/**
 * Helper function to update tag transients with post IDs
 */
function wpse317707_update_tag_transients() {
    $tags = get_tags(); // add args

    /** @var WP_Term $tag */
    foreach ( $tags as $tag ) {

        // get random image for this tag
        $args = array(
            'post_type' => 'custom_event',
            'tag__in' => array ( $tag->term_id ),
            'posts_per_page' => -1,
        );

        $query = new WP_query ( $args );
        $post_ids = wp_list_pluck( $query->posts, 'ID' );

        set_transient( $tag->slug . '-tag-posts', $post_ids, DAY_IN_SECONDS );
    }
}

add_action( 'update_tag_transients', 'wpse317707_update_tag_transients' );

/**
 * Schedule WP Cron job
 */
if ( ! wp_next_scheduled( 'update_tag_transients' ) ) {
    wp_schedule_event( time(), 'twicedaily', 'update_tag_transients' );
}

Your code, adjusted

$tags = get_tags( $args );

foreach ( $tags as $tag ) {
    // get tag link
    $tag_link = get_tag_link( $tag->term_id );

    // get post IDs
    $post_ids = get_transient( $tag->slug . '-tag-posts' );

    // get random ID
    $random_id = $post_ids[ array_rand( $post_ids, 1 ) ];

    the_post_thumbnail( $random_id );

    // display content
    echo '<div id="tag-block">'
         . $tag->name
         . $tag->description
         . '<a href="'.$tag_link.'">see tag archive</a>'
         . '</div>';
}

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