I have to say, that as a person currently in grad school, and who comes from a long line of academics I find this to be one of the most inaccurate articulations of academic training and post-doc career potentials that I have ever heard. Based on my own experience, this post extremely unidimensional in it's thought and is potentially pernicious - almost the the point of absurdity.
I can't help but feel there is some sort of agenda at work here.
Don't take my word for it though, the research here is easy to do. Simply research the individuals who currently hold the posts you'd like to eventually hold and look at their background. I can't speak for all disciplines and field, but I can tell you that, in general, there are number of within-individual factors that trump the school where you did your grad work. Essentially, the quality, usefulness, and originality of one's research are the best predictor of career trajectory.
There ARE efficacious mentors outside of top 25 programs. That's just a fact. There are people who hold important chairs at top 25 universities who did their own grad work outside of that structure - for a plethora of reasons.
Fit is more important than school rank. School rank is important as a factor and variable - but it's not the prima facie factor as this "reaistic" (read: cynic) suggests.
Good luck. Take a breath. Be yourself.