Too bad you didn't step up to the plate and become the maintainer, when Google offered to give the source code away to anyone who wanted to run their own "Google Reader" service.
It is not a problem of code, it is a problem of providing the service
When Google originally offered the code, they offered to host it on Google's hosted infrastructure service for a year, at no charge, until the project got up on its feet. There were no takers.
This will probably be moderated down as well... however, yes, "providing the service" is *exactly* the problem, and it's *exactly* why Google cancelled the thing when the back end hosting infrastructure APIs changed out from under the (unmaintained) Reader codebase. The maintainers had moved onto other projects.
And while Google could have either brought them back (the ones who wanted to revisit their old code), or they could have put new hires on the porting problem, and gotten Reader back on its feet on the new hosting infrastructure, it wouldn't have solved the basic problem.
The basic problem is that there was no sustainable revenue model for the service. Google's Reader service allowed the use of any client that someone cared to write, and a heck of a lot of people wanted to write clients that excluded advertising as a means of supporting the costs of running the service. Which would be fine, if there were any way to charge for it, *other* than advertising, which didn't break the client/back-end-service model, which is what people *liked most* about Reader in the first place.
So Google didn't throw good money after bad, and no one else stepped up to throw good money after bad, and (possibly) figure out some other way to monetize the service, such as changing the over the wire representation such that advertising was indistinguishable from content. Which wouldn't have worked, since that would just trigger an arms race for clever advertising exclusionary filtering in the display services, instead of at the protocol level.
So you're right: "it is a problem of providing the service", and the specific problem is "no one wanted to pay to do that".