If you were an editor-in-chief, would you appreciate to be informed that an associate editor has published sloppy information in his journal?
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.