RSS is still the best way to track the news on the web, and these RSS readers can keep you right up to date.
Do you need that? You only need to sync the feed. There are formats like OPML for that. At worst you need a file sync tool like syncthing. The feed contents seen by the readers are all the same.
I’m yet to see a good reason why feed readers need to be web apps. This is worse than the case of git - a decentralized tool is taken and made centralized.