I have done a bit of hacking and found the sources of the issues, and I think you really should address them.
Inside the custom file type definitions, you have archiving set to “false.” I’m not sure why you chose to do this, but it prevents sites that use landing pages from displaying paginated archives. You need to include an option in your control panel to turn regular Wordpress archiving on if the site has issues due to using a landing page.
Secondly, the posts not displaying on the landing page was indeed due to the theme’s structure, which I was able to solve. However, when pulling the excerpt for the feed, it instead pulled the full episode summary, which was a paragraph long. You need to separate the summaries from the excerpts. Perhaps the best way to do so would be to create a new database entry for the podcast summary, separate from the excerpt, with the feed RSS for the podcasts pulling from the summary field rather than the excerpt.