Your examples of the use of present simple are certainly better than those along the lines of "Birds sing", because there is a clearer reason - communication - for producing the texts (using the forms) in your examples; in direct contrast, apart from to practise (controlled and simple) form and/or learn some mightily mundane vocabulary, it is hard to think of why anyone would want to enumerate let alone hear such known (and therefore mind-numbingly boring) facts like "Birds sing". So you are right to wonder why they appear in grammars (even supposedly pedagogic ones) quite as often as they do. Is there really any excuse for it?! You have a right to be alarmed, and I share your concern.
The problem is that linguists won't be able to please everybody all of the time, and even if examples that are more authentic or more whatever can be found, it is by no means a given that students will appreciate them in the same way we (teachers, and linguists) do (see discussion re. "authentication" below).
That is, grammars (as opposed to corpora), especially for foreign learners need to take account of PRODUCTIVE needs, and it is not always easy to know exactly what those needs will be. So, unless you are (teaching) a journalist, such examples will not help you to use specific (present simple) "non-journalistic" verbs yourself (although they could well help learners appreciate what present simple "means" generally, and how it is used with "newsy" verbs specifically).
I don't think grammarians "avoid" these examples for the reason you mention (they just make bad/non-specific choices is all), interesting though your point is (in fact, I misunderstood it completely at first and wrote some nonsense - since deleted - that I hope nobody read!

): "But grammarians have seen there is a trouble. As these very common examples should be no longer said in Simple Present some days, weeks, or years later, grammar writers forcefully help teachers to explain Simple Present by not reporting them whatsoever. Instead, they carefully choose examples that may be very probably still valid in Simple Present -- as long as the book exists."
(I think that text as original reported object and text as reformulated by readers in the "real world" is not considered to be a problem, and I am not aware of many linguists who have (patronisingly, on behalf of foerign learners) seriously suggested otherwise (but please tell me more if you got the idea from somewhere or have been developing it yourself...e.g. I'd be interested to see if and how the forms used in text and reformulation differ, distort the timeframe/reference etc)).
The reason for not presenting many examples to students is therefore I think probably more to do with perceived productive needs than "avoidance"; besides, if students want to read papers there is nothing to stop them going and buying one, or consulting a (specialized) corpora! I suppose an answer is to try to achieve "balance" in the input according to what students will encounter and then need (or perhaps choose) to talk about...but this is by no means easy to achieve, as compilers of corpora have found out (see e.g. Kennedy's "An Introduction to Corpus Linguistics").
So the thing to do would seem to be for US to find examples (for productive purposes) that are more relevant and useful in OUR students' lives (which is what most good teachers will be doing already - I myself "bin" that birdy BS almost immediately) - we are in the best position to find out what students need!
A dichtomy between productive needs and reception is not necessarily helpful in the long run, however - I can appreciate that news/the media may well be of personal interest to many students - that is, such students might want to AUTHENTICATE this kind of thing in the classroom or wherever else they have contact with other English speakers more than more (to them) mundane/potentially boring things selected by a well-meaning (unambitious, even?) teacher.
My questions to you then would be: what proportion of authentic (written and spoken) news materials would you think appropriate for students to study, especially regarding their (productive) needs?
By the way, I have been reading some Widdowson by way of Seidlhofer's "Controversies in Applied Linguistics" recently and guess I want to see what people think of the whole question of "authentication" (of authentic materials) by processes of communication (as opposed to just studying?) etc.