What are the flaws of working in a research center department led by young principal investigators?
Some of the disadvantages of young professors vs. more senior staff may be:


Less established in their field. One advantage of a very senior professor for a PhD student is that the senior professor often knows all or at least most of the important players in the area. Hence, he can put the student in contact with interesting collaborators, and open up doors for internships or postdocs. A younger professor may not yet have a network that broad.
Less experienced. This one is obvious, but more senior professors are typically more experienced. They have advised many students before, and make less "beginner's mistakes" (e.g., expecting too much from a student, and then getting angry because the student did not live up to those expectations).
Less laid-back. Non-tenured professors need their students to perform, as their own career is very tightly coupled with their students (mis-)fortunes. Many a high-potential assistant professor will not take an unsuccessful research project lightly. This can be an advantage or a disadvantage - an ambitious student and a non-tenured professor may push each other to achieve great results, but the same 
professor and a more average student may lead to some friction and bad feelings.
Less money. Younger professors tend to have less research grants (yet). This has implications for you even if you personally have a stipend from another source, as it means there is less money for travel, less money to fund research students, etc.
Less pull in the faculty. This may be more relevant in some places than in others, but I have certainly seen faculties where working with an assistant professor had the disadvantage that your group was at a disadvantage whenever any sort of global resources were distributed. The senior professors were often able to acquire resources (they did not actually need) basically by appealing to their seniority.
Higher chance that your advisor moves. Non-tenured professors are much more likely to change university (voluntarily or due to not getting tenure) than ones that are already tenured. Especially take this into account if the tenure review of your professor is planned for the next years.


These things just came from the top of my head, I will edit the answer if I can think of anything else.

Sidenote: yes, these are all stark simplifications. No, none of those points has to be true for any given young professor.