Is the classical chair system (in which a senior professor supervises junior professors) obsolete?
I know of no American university in which any tenure track faculty member works under any other tenure track faculty member in any official capacity.

The main difference between assistant professors and full professors is that assistant professors do not have tenure.  (Going far enough back in the history of my department, one sees "tenured assistant professors".  I don't know what that's about.  I am not willing to claim that nowadays no American university has tenured assistant professors, but I do not know of any.)  "Associate professor" is ambiguous on this point: in most cases associate professors have tenure, but at my university the promotion and the tenure are distinct processes with slightly different rules, although they are similar enough and onerous enough that candidates get pressured -- perhaps a little unfairly, in my view -- to carry them out simultaneously.  There are a few really good places where "associate professor" is a title awarded to young faculty for which their future tenure is by no means assured -- I'm thinking of you, MIT.  But that's rare.  

You write that an assistant professor is "with a lower salary and possibilities".  I wanted to let you know that this really need not be the case.  Academic salaries are most competitive now at the assistant professor level; since annual raises have been meager or nonexistent in many recent years, an associate or even full professor cannot be counted on to have a higher salary than a new-hire assistant professor.  In fact the amount of our first offer to new-hire assistant professors in my department is very close to my current salary (I am an associate professor not so far away from promotion to full).  This means that if the candidate negotiates at all, they will get offered a higher salary than mine.  This has certainly happened.  (All this is a matter of public record.)  I think that there are no full professors in my department making less than new hire assistant professors, but there are some who are not making substantially more.   

Also, in my department and at many others, assistant professors do not have fewer "possibilities": with the exception of certain voting rights in faculty meetings, they have identical privileges with all other faculty.  They may begin with less "service responsibility" than older faculty; that is probably an advantage.