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-> get accepted by him/her -> 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?