Excerpts are not displayed by a shortcode on category pages

I have reproduced your environment using the same theme and the same plugin. Out of the box, the plugin does not register the CPT with excerpt. You need to register it:

add_post_type_support('advert', array('excerpt'));

This should go in functions.php of your (child?) theme.
My only question is: how and why were you getting excerpts on singular pages if the post type does not have excerpts enabled? I assume it’s a WP feature/bug that is not the excerpt itself, but the trimmed/stripped version of content (55 chars by default) WP builds and I am guessing it does it without checking if excerpts are enabled for that particular post type.

Alternatively, you could just add excerpt to line 108 of wpadverts/wpadverts.php, but if you do this you will lose this mod if you ever update the plugin.

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UPDATE

I think I finally understood what you asked for in the first place. My own testing of the plugin and theme is here. I do not know why the_excerpt() does not return a trimmed down version of the content on advert category pages for posts that do not have manual excerpts. But, to fix this problem, I made a “custom excerpt” for use in list-item.php. Here it is:

<div class="advert-post-excerpt">
    <span class="advert-excerpt">
        <?php
        global $post;
        $limit = 55;
        $content = strip_tags($post->post_content);
        echo has_excerpt($post->ID) ?
            $post->post_excerpt :
            ( strlen($content) > $limit ?
                substr($content, 0, $limit).'...' :
                $content
            )
        ;?>
    </span>
</div>

Change 55 to the desired excerpt length. The code above should replace your initial:

<div class="advert-post-excerpt">
    <span class="advert-excerpt"><?php echo get_the_excerpt(); ?></span>
</div>

I’m a practical guy, if it doesn’t work one way, I find a way. However, if anyone could enlighten me on why the_excerpt() returns null in this case, I’d thank them.