I agree-- RSS feeds are wonderful. I just wish more podcasts kept a complete feed. I'm a "completionist" and it turns me off when I have to scrape episode archive pages to get download links for past episodes that have dropped off the RSS feed. That's the biggest barrier for me to start listening to a new podcast.

I own an iPhone but also don't use Apple's built-in functionality unless the feature supports standards-based services can self-host (CalDAV, IMAP, etc).

I pull podcasts into my forked version of tt-rss[0] and use a script to pull down the enclosures onto my local webserver. I play the episodes using Safari (which, admittedly, is a sub-optimal experience) on my iPhone. (In my dreams I'd write an HTML5 front-end to play episodes, mark them to retain after listening, keep bookmarks, etc...)

[0] https://tt-rss.org/