Why is my taxonomy template not shown?

I have created a custom taxonomy for my site called projects with a matching template called taxonomy-projects.php. The problem is that when I browse to https://mysite.com/projects the template is not triggered, index.php is, but when I browse to https://mysite.com/projects/someterm the template shows up just fine.

How can have my template displayed when I browse to https://mysite.com/projects?

Before you ask. Yes, I’m aware that I need to clear the rewrite rules before changes are made. I do this by saving the Permalink settings in the admin.

This is how I’ve created my custom taxonomy:

$args = array(
	'public' => TRUE,
	'hierarchical' => TRUE,
	'query_var' => TRUE,
	'rewrite' => array(
		'slug' => 'projects',
		'with_front' => TRUE,
	),
	'labels' => array(
	'name' => "Categories",
	'singular_name' => "Category",
	'search_items' => "Search Categories",
		'popular_items' => "Popular Categories",
		'all_items' => "All Categories",
		'parent_item' => "Parent Category",
		'parent_item_colon' => "Parent Category:",
		'edit_item' => "Edit Category",
		'update_item' => "Update Category",
		'add_new_item' => "Add New Category",
		'new_item_name' => "New Category Name",
	),
);

register_taxonomy('projects', array('project'), $args);

I’ve traced the output of $wp_query->get_queried_object() on each page and when I browse to a page with an existing term, such as https://mysite.com/projects/commercial I get the following output:

stdClass Object
(
    [term_id] => 4
    [name] => Commercial
    [slug] => commercial
    [term_group] => 0
    [term_taxonomy_id] => 8
    [taxonomy] => projects
    [description] =>
    [parent] => 0
    [count] => 1
)

However, when I browse to https://mysite.com/projects the output is nothing, empty output.

I’m failing to understand what I’m doing wrong. How can I make the taxonomy-projects.php shown when I browse to https://mysite.com/projects?

Any help would be greatly appreciated.