Seriously, why does this question keep coming up? What am I missing?
RSS readers were useful because you had all the news/posts in one place (instead of many browser tabs/newsletters/social accounts). You read, curated and shared from the same place, and you were able to easily choose what categories you wanted to read and when.
Interesting to think that an attack vector against an open standard is creating the objectively best implementation and then shutting down after killing competition and consolidation.
Other commenters in here have mentioned their readers of choice; I can add mine: NewsBlur.
Turns out RSS is still in pretty common use, as far as I can tell, unless you can point to a handful of sites that aren't using either RSS or Atom.
Invariably, when the topic of RSS comes up, someone laments its "passing", which leads me to ask (again): where is this impression coming from? Is it the mere absence of Google Reader?
Is there a social aspect I never knew about?
I see you mentioned other tools being lacking, but I know NewsBlur, at least, provides all those capabilities. (Not sure about "curating", though; maybe you could define that a bit more? There is the "blurblog", which has its very own RSS feed, which lets you surface articles you like, and there's the "save" function.)
I still subscribe to feeds in a different tool but it's not the same.
Also, "not the same"... can you be more specific? (And which tool?)
Inoreader is the only RSS reader I'm comfortable with, Thunderbird's also ok. They aren't "widget" pages though.