I suspect an effective method could be built, and I would use such a method if it existed. However, any method would violate the sandboxing that has become vogue as we attempt to monetize the Internet. RSS didn't go away because it didn't work. RSS went away because it wasn't the most efficient way to sell advertising. Putting together a "look - this other service over there with a completely different ad network is also talking about this! Go over there!" service is a quick way to lose money.
RSS didn't really go away. Most sites still have RSS feeds, and I've never found a site I actually wanted an RSS feed for that didn't have one. RSS doesn't get promoted along with the advertiser-friendly social media panopticon anymore, but it's still there.
I use feedly which is syncd from my phone to computer. The app on Android and iPhone is great. They have good keyboard shortcuts in the browser. I got it right when Google shut down and simply logged in with my Google account - no importing necessary. It's also got a hook in IFTTT, so my saved articles are automatically added to my Evernote.