I've been hosting a fork of Tiny Tiny RSS[0] since 2005. Moving it between servers and Postgres instances has been easy and I like having the old data in perpetuity.
I don't think there is any website I care about checking regularly that does not have an rss/atom feed. All wordpress (and pretty much every other blog engine) sites do, subreddits do, https://hnrss.github.io/ for hacker news...
That said, I personally only use rss for slow-updating sites (pretty much just for personal weblogs and the news/announcements for some projects).
Quite common, most things that I try to subscribe to have a feed, including mainstream stuff like NYT, Axios, FiveThirtyEight, and smaller independent blogs. When one doesn't exist you can often use a Twitter feed instead (for the moment), but I've only had to resort to that once or twice.
Pretty common if not necessarily advertised. I'll echo what others have said-- any site I particularly care about has an RSS feed.
I've written scrapers for some things that don't expose RSS to dump into the tt-rss database, too. Having the data sitting in a relational database makes it easy to interface with. (I wrote my own "podcatcher" since 2005-era tt-rss didn't have any such functionality and just use the tt-rss database as its back-end.)
It is probably less than years ago, but its still a thing. In fact i heard youtube and other big sites (e.g. Reddit, etc.) still have rss feeds. Someone also told me that there is a bit of an uptick....buit this kind of thing is likely difficult to concretely quantify. As a single data point, I'm a fan of sites with RSS feeds...so i will frequent these sites more, and would be williung to pay for sites to have such a feature!
Yeah, when i am deciding if i wish to return to a website to read more of their stuff, the very first thing that i do is search for an rss/atom feed. If they lack such a feed, then chances are quite high that i won't return.
[0] https://tt-rss.org/