I'm sure someone from Google will tell you it is to control the quality/experience. I remember back when I used iTunes that sometimes one podcast or another wouldn't download for a while, because their server was down or whatever, whereas most Google things it is reliably never the server's fault. Google would not be wrong to say it can ensure a better user experience if its providing the podcast over referring to another server which could very well be someone's Raspberry Pi.

But to be listed on Google Podcasts, they consume your RSS feed and upload it. The terms include the standard granting Google irrevocable worldwide royalty-free rights to your intellectual property and includes highlights like their ability to use your podcast content even if you remove it from your feed later.

Since most other podcast apps don't do anything but help others find your RSS feed, I've never had to agree to terms and prove my ownership to be listed anywhere else, Google's demand of IP rights seems unique in this space. I've actually found my podcast listed on all sorts of sites I've never heard of with no such terms required.

Sounds like AMP. "Google hosts it for performance reasons!"

I'm not too worried for myself since I plan to release it all in a Creative Commons album anyway, but I'll keep it in mind if I see someone talking up Google Podcasts.