I don't know with any certainty if the deprecation of shared hosting support has anything to do with some weirdness I'm experiencing on the shared hosting account where I have my TT-RSS installed. So, I am preparing for the worst.
I want a self-hostable and FOSS solution if at all possible. The solution also needs to play well with mobile (Android), either through really good Web site design or a dedicated app. I also insist on keystroke article navigation (next and previous).
I'll update this list as I try things.
FOSSownCloud sports a news module. I'm taking it for a spin through a free as in beer provider. The desktop experience is fine--apart from the ordering of articles, which is almost chronological. The mobile Web experience is nearly tolerable. There is a dedicated News app, the usability of which is pretty good, but it has some arcane sync setting configs to get it to sync with the "main" (i.e., Web) list of read articles. You'd think it would do that by default.
FreshRSS looks promising, but I am having issues with the app's not marking read articles as read. I don't know if this is a problem with my provider's DB connection. (It's not the same provider as my TT-RSS install). Keyboard navigation is fine, and the mobile Web experience is workable. It's claimed that the EasyRSS app can connect to FreshRSS, but I have yet to try it. The desktop web interface also has what I feel is a significant usability issue: if you set articles to me marked as read on scroll, then when you scroll the page to select the category of feed you want to read, you manage to mark whatever articles you scrolled through at the same time as read.
Proprietary hostedAOL Reader comes as a bit of a shock: it's quite decent. Has the needed features and the mobile Web experience is tolerable. It has a 1000 feed limit.
G2Reader also comes highly recommended. But it lacks keyboard navigation, so poop.
Digg Reader looks promising, but I don't like that the only ways in are via Twitter ("Read Tweets from your timeline. See who you follow."), Google ("View your email address, View your basic profile info, Manage your data in Google Reader"), and Facebook (no thanks).
Feedreader doesn't seem to get a lot of ongoing love from its developers, with the most recent news being from 2015 and 2013. This makes me think it could go away without much notice.
The Old Reader has a 100 feed limit for free accounts.
NewsBlur has a 64 feed limit for free accounts.