Wordpress - Display all posts in a custom post type, grouped by a custom taxonomy

So, you might consider automating the multiple queries.

First, get the list of terms in your custom taxonomy, using get_terms():

<?php
$member_group_terms = get_terms( 'member_group' );
?>

Then, loop through each one, running a new query each time:

<?php
foreach ( $member_group_terms as $member_group_term ) {
    $member_group_query = new WP_Query( array(
        'post_type' => 'member',
        'tax_query' => array(
            array(
                'taxonomy' => 'member_group',
                'field' => 'slug',
                'terms' => array( $member_group_term->slug ),
                'operator' => 'IN'
            )
        )
    ) );
    ?>
    <h2><?php echo $member_group_term->name; ?></h2>
    <ul>
    <?php
    if ( $member_group_query->have_posts() ) : while ( $member_group_query->have_posts() ) : $member_group_query->the_post(); ?>
        <li><?php echo the_title(); ?></li>
    <?php endwhile; endif; ?>
    </ul>
    <?php
    // Reset things, for good measure
    $member_group_query = null;
    wp_reset_postdata();
}
?>

I can't see anything particularly wrong with this approach, though it may have a limited ability to scale (i.e. if you have hundreds or thousands of members, or member_group terms, you may see performance issues).

I found a solution by using a custom query and then grouping it with the term name:

SELECT * 
FROM wp_term_taxonomy AS cat_term_taxonomy
INNER JOIN wp_terms AS cat_terms ON cat_term_taxonomy.term_id = cat_terms.term_id
INNER JOIN wp_term_relationships AS cat_term_relationships ON cat_term_taxonomy.term_taxonomy_id = cat_term_relationships.term_taxonomy_id
INNER JOIN wp_posts AS cat_posts ON cat_term_relationships.object_id = cat_posts.ID
INNER JOIN wp_postmeta AS meta ON cat_posts.ID = meta.post_id
WHERE cat_posts.post_status =  'publish'
AND meta.meta_key =  'active'
AND meta.meta_value =  'active'
AND cat_posts.post_type =  'member'
AND cat_term_taxonomy.taxonomy =  'member_groups'

Then by just using a regular foreach query I can just extract the information I want.

But I'm still interested in another way if there is, maybe by using Wordpress' own functions.

even simpler:

$terms = get_terms('tax_name');
$posts = array();
foreach ( $terms as $term ) {
    $posts[$term->name] = get_posts(array( 'posts_per_page' => -1, 'post_type' => 'post_type', 'tax_name' => $term->name ));
}

Within the resultant $posts array, each tax term is the key to a nested array containing its posts.