How to Set Up Podcast Custom Post Type to Send to iTunes

I created a custom post type called Podcasts, and in it I have a custom meta box generated by the Advanced Custom Fields plugin where a user can upload an audio file. Then, on the archive page and single post pages for the Podcast type, I’ve placed the output of that custom meta box within an audio tag like so:

<audio controls>
<source src="<?php the_field('podcast'); ?>" type="audio/mpeg">
Your browser does not support the audio element.
</audio>

The audio files are showing up fine and playing fine. So my next step is to hook up the feed to iTunes. I would like to do this without using a plugin, if possible.

I followed the instructions here to set up a Feedburner, but when I tested it in iTunes, nothing showed up. It said there were no podcasts in the feed (even though there are 2 posts in the archive with audio). What am I doing wrong?

Here is a link to the archive for the custom post type that I am trying to work with.