How to tackle the bibliographic work as an entering PhD student?
To be honest, it sometimes feels like a sneaky way of procrastinating:
  I'd tell myself that I'm still not ready to tackle my problem and that
  I need to make a little "detour" by existing works; and when it's done
  I don't feel more at ease to answer my questions. Or is it a normal
  aspect of research?


It is probably the most common way that new grad students attempt (consciously or otherwise) to avoid doing research :)

You have to remember that prior knowledge is an ocean you're swimming in, not a cup that you have to finish drinking. There's literally no way to "read all the prior literature": there's always something more. 

General reading is perfectly fine, but when you have a problem to work on, it's best to break down bibliographic research into:


what does prior work say about your problem and what's the key open question (and this is something your advisor can help you with)
Once you start thinking about a specific problem and are trying to apply some specific tools, is there material to help you understand those tools better ? 
When you're writing up, what related (not prior) work should you be aware of ? 


At each stage, read only as much as you absolutely need to, in order to avoid going down rabbit holes. And remember, as a new Ph.D student, it's not a terrible thing if you reinvent something that someone discovered before, and only learn of this after the fact. You just don't that happening at your dissertation defense :)