One piece of anecdata: I subscribed to the Economist briefly last year and it was one of the best things I've done as far as advancing myself as a person. While the vast majority of what written about wasn't immediately relevant to me (compared even to a random programming-related blog post), it did a lot to enlighten me on the value of good journalism and broaden my world view as well as keeping me reasonably informed on happenings in the world. I stopped subscribing to the Economist because I couldn't find time to read every edition cover to cover, but I plan on starting again this year.
I also subscribe to https://lwn.net which brings up another thing -- supporting the organizations that make the news is important if you want news that is good (even if it's irrelevant right now) from being produced.
I am an Economist subscriber and just download the audio edition each week. I listen to it when doing other tasks that don't require 100% mental focus such as housework and am usually able to complete each week's issue before the next one releases. I am not absorbing the content as well as if I were to sit down and read each issue, but this is a reasonable tradeoff for me.
They also have an android/iphone app for subscribers, but I've had a lot of issues with playing audio through it and find it much easier to simply download each week's .zip file of .mp3's and play them with my audio player of choice.
It is hard to find time to read The Economist from cover to cover. I try to at least look at each article, and read at least half of them. Even The Economist has some articles that seem ephemeral: for example, a discussion of whether a particular merger will or won't happen.
I have had their subscription twice and the only real complaint for me would be, as you also touch upon, that they almost seem to come too often. If only they had a cheaper "digital only" subscription.
I also subscribe to https://lwn.net which brings up another thing -- supporting the organizations that make the news is important if you want news that is good (even if it's irrelevant right now) from being produced.