If you want to take it a step further, add some links feed readers that run as browser plugins. Unlike hosted services, they're not as likely to disappear, and they have a low barrier to entry since they don't require any sign up.
I've been working on one that runs in Firefox[1], but there are many others that look promising for Chrome as well.
You mention a lack of hosting but would it, optionally, browser-sync?
I use a web interface, The Old Reader, to access feeds from 4 devices. Whichever I'm using at the time knows which articles have been marked as read; without using a different client for each OS platform.
So a browser extension that syncs to one's Firefox account, like the bookmarks/history feature would be handy, with no additional hosted service other than having a browser login.
Yeah that's feasible, I haven't experimented with it yet.
Personally, I find that I read very different things on my laptop than I do on my phone (technical articles vs. news) so I haven't had a huge motivation to investigate it, but it's an interesting avenue. The only possible stumbling block is that the size of sync'd data is relatively constrained, about 200k I think, which is probably enough to hold a user's feed list and track whether they've viewed/ignored various articles, but it would require some planning.
Firefox's native support going away doesn't really make a big difference, as far as I know you couldn't access it from a plugin anyways.
There are many parsers that are adequate, especially if you don't try to get too fancy. If you're curious, the source is available on github.
https://github.com/adamsanderson/brook
I've been working on one that runs in Firefox[1], but there are many others that look promising for Chrome as well.
[1] https://addons.mozilla.org/en-US/firefox/addon/brook-feed-re...