From hearing the developer of Overcast talk about this issue, it seems that hiding your IP is really all you can or need to do, as that’s what the dynamic ad services use to track you

Overcast itself is fine to my knowledge but it doesn’t do anything on its own to protect your IP from getting to the podcast hosting servers, which do all sorts of creepy shit. The only real solution would be to tunnel your downloads through some kind of VPN or centralized download service, and those aren’t services Overcast provides so you’d have to do it yourself.

ohhh that’s a good use of it. definitely not something i’d have thought of trying