Only Show Excerpt After First 3 Posts

You could add a count variable and count how many times the page has looped and only show the excerpt if the count is 3 or greater.:

<?php

    $Count = 0;
?>
<div class="content-block-archive">

     <div class="content-block-archive-thumbnail">
<a href="https://wordpress.stackexchange.com/questions/343700/<?php the_permalink(); ?>"><?php echo get_the_post_thumbnail($post_id, 'large', array('class' => 'alignleft')); ?></a>
     </div>

     <div class="content-block-archive-meta">
<h2 class="content-block-archive-title"><a href="https://wordpress.stackexchange.com/questions/343700/<?php the_permalink(); ?>"><?php the_title(); ?></a></h2>

          <div class="content-block-archive-author"> <?php the_author_posts_link(); ?> | <?php echo get_the_date(); ?> 
          </div>
        <?php 
        if ($Count >= 3) { ?>
            <div class="content-block-archive-excerpt">
                <?php 
                    the_excerpt(); 
                ?>
            </div>
        <?php 
        } 
        $Count ++;
        ?>
    </div>
</div>