I see. Do you know if any other browsers handle this situation better? That might carry some weight for a feature request.
A custom extension might be a workaround. It could autofill the username based on the path or login form. Or, it could at least add something to the page showing you want site (path) you're on as a reminder.
Feedly upgraded their login system this week to support more types of logins. The above procedure will need to be redone, but it all still applies with the exception of one thing: the cookie's name is now "feedly.session" instead of "session@cloud," though YMMV if you're using some of the new types of logins, in which case take note of the name when you're retrieving the cookie from another browser.
The "solution" I've taken to is leaving JS disabled on nytimes.com and various sub-domains.
Still, if anyone knows of something less Draconian (perhaps including just the right thing in urlfilter.ini), it would be great to know. Most pages are perfectly readable without JS, but you miss all the extras like slideshows.