I appreciate the distinction you make between urgent and non-urgent news.
Finding out about things quickly isn't necessarily my priority. In fact, one of my problems with "regular" news outlets is that they have poor sense of time sensitivity, and promote news that's stopped being useful. The value of knowing about Icelandic volcanoes grounding all northern European air traffic is actually very useful to me when it's just happened, but in a week's time I may as well read about it on Wikipedia.
I'm more concerned about finding out about things at all. My ad hoc news accretion drops the ball more often than I'd like. My ideal wish-upon-a-star would be a daily digest saying "here are a list of things that have happened today in two sentences or less". I can then decide whether to follow it up or not.
(I have a secondary motive of wanting to associate events in my memory to improve the granularity of my recall. I know, for example, that Eyjafjallajökull erupting was concurrent with the run-up to the 2010 UK General Election, which helps me position it in time quite accurately, as well as position personal events that I remember happening around the same time.)
Hilariously, a good option for you may be an actual newspaper. Made out of paper.
It comes once a day, it summarizes a few dozen major events in a reasonably succinct way, and many of them try to minimize reporting bias. You could consider specific papers based on size and editorial style (most offer free or cheap trials), and then sign up for a short subscription to see how you like it.
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