To my mind, one of the big problems of training new researchers is an institutional insistence on treating trainees like children. This attitude, as much as anything else, is what leads people to be so thoroughly unprepared for the leap to assistant professor.
There are certain responsibilities that a PI has that no graduate student or postdoc can ever fully experience. Hiring decisions are one. Managing the budget of an entire lab is another. The balance that must be maintained in order to train postdocs and graduate students in the practice of science is such that they cannot devote as much energy to the higher level or managerial tasks as the PI can/should. However, these tasks and experiences can and should be introduced to trainees judiciously. Postdocs and graduate students can review hiring dossiers and be on interview panels. Trainees can be given control of portions of the budget relevant to their projects. Better, they can be encouraged to apply for grants themselves. Many PIs in fact do this. My current mentor is wonderful at including me in as many aspects of running her lab as she can, explicitly so that I am exposed to the realities of lab management before I (hopefully) get to start my own lab.
Yet, at an institutional level, there are barriers to allowing trainees to begin developing all the management skills required of a PI. The story that follows is from my own graduate student years. It still irritates me.
Before I started graduate school, I worked in the real world, such as it is, for a couple of years. (As an aside, this is something that I think is invaluable in a graduate student. Anyone who comes back to academia after having experienced the world of employment knows why they're there). For six months in that period, I was part of a three person development team for a science communication charity on a major fundraising drive. In that time, we developed a strategy which included approaching new donors from all three major branches: corporate, individual philanthropy and foundations. I ended up specializing in foundations, and coordinated the writing of several grants. A few were awarded, and I raised my own salary several times over. In development, this is the basic benchmark of how good you are at your job, and whether you're worth what you're being paid.
A few years later, as a graduate student, the scientific society of which I was then a member sent out a request for new members of the fundraising board. I sent out an email detailing my experience in fundraising and offering to serve on the committee. Although the head of the committee was initially enthusiastic, the society did not allow graduate students to serve on the committee. To this day, I cannot see how such a policy can benefit an organization when it overlooks the fact that I had directly relevant professional experience. Only a mindset that sees graduate students as intrinsically less valuable, rather than as young adults with several tears experience, could defend it. In putting up this barrier, I was deprived of valuable service experience, and the society was potentially deprived of my experience in the topic.
It's entirely possible that my experience was not unique, in which case my lack of experience would certainly count against me. This was not the reason I wasn't allowed on the committee. So before saying that trainees are not prepared to be assistant professors, ask yourself: are there any barriers to trainee involvement in more managerial tasks that exist solely because of their trainee status, and not because of any fundamental lack of ability? And how can we find ways to include trainees more in these tasks?