I use RSS a fair amount. I can't really speak to the accessibility, but I find feeds by searching a web page's source for the word "feed". The New York Times has feeds available from this page for different locations and topics.

I use and recommend a selfhosted FreshRSS instance, but I'm unsure whether or not it plays nice in terms of accessibility.

For finding feeds I mostly just look for RSS on websites that I come across. Another thing I did when moving to FreshRSS was using Feedly, a paid RSS reader with a free plan that I was using, to find some interesting sources and import those into FreshRSS. I also use Reddit RSS feeds for a few subs that don't have a big enough audience outside of Reddit.

Unread articles is no problem for me. I just have a very quick glance and then mark everything as read if there's too much. After that I check it a few times per day, only check what seems interesting, and mark the rest as read again.

@RareBird15 @mastoblind @main
As it happens, I'm working on an rss feed discovery tool, cleverly called... discover 🤔 It has basic screen reader support but I've not done extensive testing. I'd love any feedback you have!
https://discover.brine.dev/