the_excerpt not showing posts

With a quick glance at the code, it looks like you are using the global $wp_query, but you haven’t declared it. Before you can use this you would need to declare:
global $wp_query;
I’m not sure why you even have this bit in your, code though. Your logic looks like it’s trying to check if there is only one post, then show the excerpt if so, then you go and show the excerpt on else anyway.

Can you give a more detailed explanation of what you are trying to accomplish?

I have a landing page that will, among other things, feature a small excerpt of the most recent blog post.
Currently, that section has only a link to the landing page.

You will probably want to do something like this:

<?php

// Get the most recent post
$the_query = new WP_Query( 'posts_per_page=1' );

// Pull the excerpt
while ( $the_query->have_posts() )?: $the_query->the_post();
	the_excerpt();
endwhile;

// Reset Post Data
wp_reset_postdata();

That works perfectly, thank you.