I've attempted and investigated but can't take action for this. Essentially I am attempting to display a specific publish (publish_id 1) around the home page using its featured image.

The title and excerpt work fine, however the featured image doesn't.

<?php
$post_id = 1;
$queried_post = get_post($post_id);
$title = $queried_post->post_title;
echo "<h2>" . $title . "</h2>";
echo "<p>" . $queried_post->post_excerpt . "</p>";
the_post_thumbnail();
?>

Any suggestions could be useful.

Try changing:

the_post_thumbnail();

with:

echo get_the_post_thumbnail($post_id);