Strategy for applying to PhD in the US My question was originally designed to be very related to this one.
However, I have continued to write it because it has a second part. 

I am currently sending email to PIs in order to know if they have any position available for doing research in their lab (life sciences). My list includes some elite universities but it has a wide range (top 100 universities according to shangai rank). A problem I have is that I'm interested mostly on PIs and not so much on university X or Y (every university of the list would represent a major step forward on my career). I haven't had any repply so far but I'm still optimistic.

Reading some lab's webpages I've found messages like these on contact pages.


  If you are not graduate student (...), the most direct way would be to
  apply to (INSERT PHD PROGRAM HERE).


Which is very related to


  Please be aware that individual researchers have not influence on
  admissions comittee (...)


But also this, 


  Please contact me if you would like to discuss our graduate program or
  the research in my lab.


I know that mostly in Europe the logic for Msc or equivalent that rules is

mail to PI-&gt; get accepted by him/her -&gt; apply to university with very high chance


The thing that I like about that is that it doesn't matter that a particular university has only one place that you might be interested in. However, it may be useless to apply to a university where you have only one "awesome lab" meeting your interest and more wise to apply to a university where you have (say) 3 "OK labs". Although maybe "you having thumbs up from a PI" might change the strategy, several answers point out the variability on admission comittees and the fact that your applying to the university first (you have a strong coursework component during the first 2 years) definitely make the advisor's word less important. 

So the question is: If PIs are less involved, wouldn't be a better strategy to apply to the university where there are more labs without contact or does the "european" logic still hold in the US?
My acceptance to my PhD program was a bit outside this model since I had already done some undergraduate research with my advisor, nevertheless, I think I can fill in a few gaps. 

No matter what you do, you will have to apply to a PhD program not a PI directly. For me, at my university, that meant using the university-wide online admissions process but targeting a particular college (the College of Engineering) and major (Aerospace Engineering). There was one process with customizations for each college and department since each had different requirements and requested materials. You generally can't avoid this process. Since these applications are tailored to the college and department, the admissions committees are at the departmental level. You don't necessarily apply directly to a department, but you materials will be routed there. Some universities may use separate processes for each department, but I haven't experienced that. Nevertheless, the decision will most probably be made at the department level.

Now, in parallel with your application, if you can establish a PI that you want to work with and who wants to work with you, you should do that. You may be accepted by the university before or after you are able to establish a relationship with a PI. That's OK. I suspect, though I don't have any direct evidence since I have not sat on any admissions committees, that if you do have a PI that wants to work with you before your application is considered, then they can weigh in on your behalf when the admissions committee meets. Having someone to stand up and say "I have funding and will support this student" goes a long way towards admissions for borderline candidates. For top candidates, it probably doesn't affect the admissions chances much, but it does help the department know in advance that finding them a supervisor will be easy. For borderline candidates, it can make all the difference. 

I know (second hand) of at least one candidate that was admitted on their second or third try only after they came and worked for a research institute on our campus and established a relationship with a PI. They were both able to find a backer in the committee meeting and spend some time writing a few papers to improve their CV. This latter model is unusual, but they were determined to get a PhD from our school.