One can certainly consume news without entering a state of permanent anxiety. If one does, I think that suggests an independent condition rather than a problem with the news. A healthy stoicism does not demand ignorance.
With a great deal of goal-directed effort, an individual could get there, but realistically getting our existing population to effectively adopt any meaningful level of reduced emotional reactiveness to these triggers would require a massive culture shift and billions of billions of hours of therapy and related activities. In other words, it's not a realistic short- or medium-term position at the population level, so it's at best only a small part of any solution.
This isn't to say that finding the news stressful is some sort of moral failing. It's irrational, but humans are irrational. If reading the news stresses you out, then sure, don't read the news.
Personally I can't remember the last time I experienced anxiety reading the news - and I read an hour or two's worth daily. So I have trouble relating to those who do, but I don't insist that they must be "underestimating" themselves.
Is being “well informed” on current affairs worth that much?
Seems odd to question the value of reading news, considering.