If you were an editor-in-chief, would you appreciate to be informed that an associate editor has published sloppy information in his journal? (Edited with thanks after the early comments helped gauge the perception of the post.)

This is the situation

I came across an article published in a reputable journal in the field of physics and engineering science. 


The first author of this article is a member of the editorial board.
A handful of other co-authors follows. They are also affiliated to serious education establishments and hold positions with some standing.
Author and co-authors are associated by a grand research project. This lets me think that this article attracts and will attract attention for a while, at a minimum via legit self-citations.
Parts of the article had been copied and pasted, or slightly edited, from grey literature related to the same research project. 
One of these parts is relevant for understanding the inner workings of their research outcome and, pragmatically, comparing my work and theirs. But


some terminology is plainly wrong (apples-for-pears)
some terminology is subtly wrong (community-bound misnomers)
some formulas lack
the references do not contain information integrating these lacunas


Therefore, it is impossible for me to verify the work they did and take at face value some claims of the article relevant for my investigation.
I have reached out to the research team 


first asking for additional information (I had started my study from the item of grey literature of point 4)
the first author acknowledged the apples-for-pears glitch but did not provide the lacking information
I asked again for that integration I needed
radio silence
I then discovered the article, pointed out to the inconsistency with the paper, and urged the need for an integration, out of lack of alternative sources
radio silence 
I moved on, carried an analysis of the topic at a considerable time overhead, and proposed to their judgement an integration for the lacking parts
radio silence



To sum up, 1-3 describe the context; 4-5 describe the problem; 6 describes what I did in search for clarifications.

Now onto the considerations

The shortcomings I have spotted must have passed several coarse filters.
The number of eyeballs (several co-authors and perhaps a couple of reviewers) that could/should have spotted at least some of these shortcomings from the outset is high. 

I feel that the situation is particularly anomalous because of point 1.
That is, granted none is perfect, I would expect that associate editors are particularly strict ensuring that their own publications set a standard for the journal, or do not betray that standard. 
In my view, this is surely no research misconduct, rather either a questionable research practice or a minor shortcoming, depending on judgement.
However, I feel this is not exactly OK.

I am also quite willing to wait for a few weeks for a satisfactory answer to come or for me to develop another perspective on the state of the play, also thanks to this community.

Now onto the questions

In this meantime, at any rate, I would like to gather views from this community on why


you would recommend to inform the editor-in-chief of the shortcomings in points 4-5 and motivate a request for revision?
you would rather recommend not to do this?


(I am neither an associate editor or an editor-in-chief.)
Reading your post, I did not understand exactly what the problem is with their article. It may be sloppily written, but your post comes across like you are looking for reasons to complain about it.

It is unfortunately common for articles to be poorly written. Of course, depending on the journal one expects varying quality, but even in top journals writing and presentation quality may not the primary concern for publication.

On the other hand, if you have comments on how an article can be improved, you have to be very careful when emailing the authors. Unsolicited criticism does not usually go over well.


  
  A handful of other co-authors follows. They are also affiliated to serious education establishments and hold positions with some standing.
  


This does not seem relevant to the question.


  
  Author and co-authors are associated by a grand research project.
  


Similarly, this doesn't seem to be relevant.


  
  Parts of the article have been copied and pasted, or slightly edited, from grey literature related to the same research project.
  


Are you accusing them of plagiarism, or did they only copy from their own past papers? If the former, that is a major concern and you should contact the editor, but only about this particular point, not the rest. If the latter, note that attitudes on self-plagiarism vary and while it is inadvisable, it is not always considered unethical.


  
  some terminology is plainly wrong (apples-for-pears)
  some terminology is subtly wrong (community-bound misnomers)
  some formulas lack
  the references do not contain information integrating these lacunas  
  


These are issues for the presentation but not issues with the work itself. It sounds like you have a lot of ideas for how to improve the article, but as you are not the author of the article, there is no direct and easy way for you to do so. If you feel strongly about presenting their work more effectively, you could consider writing a blog post or other piece of exposition putting their work in a positive light and clarifying such issues.


  
  I asked again for that integration
  radio silence
  


Were you polite in your emails? Did you compliment their work at all? Did you express interest?

If not, I would probably respond with radio silence too. Yes they could improve the article, but they did not ask you for your unsolicited feedback. They are not obligated to answer your email.


  I then discovered the article, pointed out to the inconsistency with the paper, and urged for some integration


It sounds like you emailed the authors and just told them to change their paper. This comes across as rude. You need to be more tactful in your suggestions.