This.

My preferred general news programme is On the Media, a weekly broadcast (and podcast) by WNYC Studios, nominally examining the media itself, almost always in the context of current news, or some background with relevance to current news.

The show runs an "NPR hour" (about 51 minutes), and covers four stories, devoting about 15 minutes to each. It's generally finalised on Friday, and broadcast Saturday / Sunday in most markets. Segments based on an interview with a subject-matter expert, rarely a typical bloviating talking head, occasionally someone directly involved in a story. The interviews are closely edited.

The result is a concise, thoughtful, informed summary of major events of the past week, without the drama, emotion, and raw churn of the usual 24/7 news cycle. Yes, you'll miss out on a bunch of stories, the vast majority of which don't matter.

The hosts, Brooke Gladstone and Bob Garfield, are excellent, with a biting dry humour. Brooke's "mmm-hmm"s can speak volumes.

The biggest ommission is whatever local news is relevant to you. Unfortunately, that's a news hole that's not being filled by much of anyone at the moment, outside a very few large markets (NYC, Washington, London, in English-language markets).

I'll also take an occasional scan of centrist headlines (Reuters of late), which tends to be very solidly balanced. Financial Times has a good rundown as well. Note that I try to restrict this scan to just headlines.

For complex and evolving stories -- the current coronavirius outbreak is an example, though the first case I'd noted was the 2004 Boxing Day Indian Ocean earthquake and tsunami -- Wikipedia articles make for excellent ongoing summaries of best current information. Vastly superior to virtually any news outlet. (Brad Plumer's Oroville Dam coverage at Vox being a notable exception: https://news.ycombinator.com/item?id=16538941)

20 years ago, strictly, before the Elian Gonzales story, I'd have suggested NPR's news magazines (Morning Edition, All Things Considered) as well worth the time. I no longer feel this, and haven't for a well over a decade. When NPR was more a suffering network without realtime capabilities, stories were submitted, edited, prepared, and largely rolled from tape. The 9/11 attacks made the network aware of its "dark secret" that much of the news programme was in fact pre-recorded. Whilst a negative in the case of critical breaking news, for the most part I'd argue this was a hidden strength of the network, as it had time to at least somewhat digest, consider, and edit what went out on the air. In the past decade, NPR (and other national broadcasters: CBC, BBC, ABC/Australia) have leaned far more on live reports on-air ... leading to worse reporting, far more technical glitches (often in the midst of interviews), and a generally more annoying product. The notion of inserting a slight delay -- 5 minutes even, though an hour or day generally wouldn't hurt -- in the "magazine" coverage would hugely improve the end result.

This is apparently an unpopular opinion.

It's also possible I've simply learned enough to see the holes in much coverage by these outlets.

And I no longer find the urge to listen to news on even a daily basis, as you suggest.

Replacements include several podcasts, notably the back-catalogs of "The History of Philosophy Without Any Gaps", "The New Books Network", and "LSE Lectures", all of which present thoughts of experts, distanced from contemporary hubub, at length, and though the direct relationships are reduced, the relevance to what's happening now is often great, with tremendous insight.

An abundance of information means a surfeit of that which inforation consumes: attention. (I believe that's Herbert Simon.) Information overload is really filter failure (Clay Shirky), and a crisis or catastrophe is a time in which previous models of the world no longer fit the present reality (my own notion, though with similarities to numerous others' observations).

We're in the midst of one such catastrophe -- a period of revelation, not in the religious sense, but in the way of a falling away of scales of no-longer-appropriate models.

(This is one of the problems with news coverage as journalism attempts to fit current events into pre-existing narratives -- when those narratives are false, the reportage similarly fails.)