p-values in sign test If we are given a list of $n$ independent measurments $\{0,0,1,0,1, \dots, 0,1,1,0,1\}$ of binomially distributed random variable $X$, we can use the sign test to check, whether the difference between numbers of $1$s and $0$s is statistically significant, i.e., whether we can reject the null hypothesis that the parameter $p$ in $\text{Binomial}(n, p)$ equals $1/2$.
Since the distribution (under the null hypothesis) of $X$ is known and the probabilities $P(|X - pn|>k)$ are not hard to compute using computer,

why it is so often mentioned that z-test can be used for obtaining p-values in the sign test (when $n$ is big enough), while, at the same time, the exact procedure is not even described?

 A: Not all books do this, of course -- many books specifically for statistics students, for example, discuss this properly.
However, you're correct that a lot of basic books designed for students in various application areas do this. (They also tend to have a long list of other strange flaws.)
[I'll avoid mentioning some of those application areas by name here. But you also sometimes see it in books written by some statisticians for general introductory statistics subjects]
In effect, you're asking us to speculate why authors of some elementary books do certain things, which is likely to reduce your question to an opinion-based one unless some of the authors concerned turn up and explain why. Nevertheless I'll make an attempt to offer some explanation.
From observation over several "generations" of textbooks*  I think there are 
* I suppose I'd consider a textbook generation to be a bit under a decade -- about the time it often takes for a widely used "standard" book to be replaced by new one rather than using another edition of the old one. 
The real answers will vary from book to book. Here are two explanations that I am quite confident cover some of the cases I've observed.


*

*A lot of the time no thought at all will have gone into this - it will be presented as a z-test because that's what was in the textbook the author used when they did statistics. 
[I often work in a particular application area that does a similar thing with other statistical issues (though they get the binomial/sign test right), simply rehashing what the previous generation did, as they in turn copied what their own predecessors did without any real thought as to why the original choices were made that way back in the first half of last century. In that case it's very interesting to trace back and find the original papers some of the traditions taught to students have grown from - many of the reasons for those choices were no longer relevant by about 30 years ago. The authors of those papers would presumably scorn the lack of scholarship that followed them.]

*Occasionally some authors of elementary methods books will deliberately choose to avoid teaching the binomial distribution** and thereby go straight to normal approximation. 
[I'm not sure this is necessarily a good choice (I think the binomial is easier to understand than the normal), but you can't put everything into an elementary methods course, so it might at least be argued to be reasonable in that case.
**(though that may happen less since logistic regression has become more widely used -- people may want to mention the binomial earlier) 
