I self host the open source FreshRSS[0] and on iPhone connect to it using Reeder[1]. Reeder does support subscribing directly to RSS feeds from the phone, you don't have to run something like FreshRSS to use it.

0: https://www.freshrss.org/

1: https://www.reederapp.com/

Thanks! Reeder is a blast from the past! :)

I can't find any info on whether or how well it handles video feeds and podcasts. Do you use it for either of those things?

I don't use it for that, but a quick test showed that with podcasts it will continue to play them if you switch to another app, and the media playback control does appear in the iOS notification centre. But it doesn't do anything clever like persist your playback position, or let you navigate around the app while continuing to listen to a podcast. So not that great really :(

I use readkit + inoreader