Hi marra8388
welcome to the Apple (user-to-user) Community
Re: My podcasts are not downloading, or playing if not downloaded
Wondering which iOS version is installed on your iPhone 13 Pro.
If you haven't already, updating to the latest version may help
Eg: About iOS 16 Updates - Apple Support says iOS 16.5 is the latest version
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Find out if you can use Apple Podcasts, and other Apple media, in your country or region
Availability of Apple Media Services - Apple Support
You might also want to check what settings you have on / off: in Settings > Podcasts
Seems any downloaded podcast episode should appear in the Library.
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Else:
Wondering if Apple support's articles (links below) can help download an episode:
Download and delete episodes in Apple Podcasts on iPhone and iPad - Apple Support
Also
Download, save, and share podcast episodes on iPhone - Apple Support
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More trouble-shooting if needed:
Does your iPhone have enough memory / storage space free (eg at least 8 Gb) in order to operate smoothly ?
How to check the storage on your iPhone and iPad - Apple Support
If needed, see: "Manage space on your iPhone, iPad, or iPod touch" in the article below:
If you need more space for an update on your iPhone, iPad, or iPod touch - Apple Support
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What to do:
If an app on your iPhone or iPad stops responding, closes unexpectedly, or won’t open - Apple Support
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Else: If you want to: you can send feedback to Apple:
Web-form for feedback
Feedback - Apple Podcasts - Apple
All the best :-)