Besides the naysaying, I think this thread has sparked a really good conversation! I'd be interested if anyone else has other ideas besides the "internship" idea -- which I think is a really good one.
My alma mater, Johns Hopkins School of Medicine, has a budding PhD student internship program (http://bci.jhmi.edu/). (Full disclosure: I pushed hard for this while a student and gave advice when it was starting.) I'm sure there are others throughout the US, and plenty in Europe as I understand.
Though it doesn't take the talent shortage problem head-on, it does provide that much-needed first step into non-academic careers like bio/pharma and policy. The hardest part was (and remains) convincing others to change the status quo. Academics need convincing that this is a worthwhile endeavor for their students, and to a lesser extent companies need to develop infrastructure to make it worthwhile (in my experience, almost no company objects to having interns around.)
Perhaps another layer of training would help, too: explaining to students how to get the most out of their short time in an internship. That's networking, doing many informational interviews and finding a foothold in a new industry.
Basically, getting everyone on-board that this will help would go a really, really long way. But it looks like we're starting, at least!