I think the crux of the issue is captured in this one statement: "Every year, another 10,000 biomedical and clinical PhDs join the ranks of postdoctoral fellows, while funding and faculty positions flatline."
In medicine, the competitive specialties are "gated" at the door. This means that it's competitive in the sense that few people actually get through the "doors" of the specialty and instead have to do something else. In science, the competition is beyond the door. It's gated at the level of getting grant funding.
Now this creates a huge issue. In medicine, while there's intense competition to get through the door of competitive specialties, once you're in, you're in and can be guaranteed to become a doc in that specialty with all its benefits. If you don't get in or are not competitive enough, then you end up doing something else. You still become a doctor in whatever other specialty you go into, with all of its benefits. In other words, you don't end up training for 5+ years in a competitive specialty only to find out that you may never be hired to do that job. The problem with science is that the gating mechanism is at the grant level. So you get your PhD, but that does not guarantee you a job or funding. So now you've spent 5+ years of your life training for something that you may never end up doing. This creates an increasing population of PhD scientists who cannot capitalize on their PhDs. You don't have the same phenomenon with MDs.
So the solution, in my opinion, is to move the gating mechanism in science back. Graduate fewer PhDs who are of higher quality. Now, there are multiple challenges to this, including the fact that science is driven by high-throughput publication. PIs are rewarded for quantity (and to a lesser extent, quality) of their work. So it's to their advantage to hire as many grad students as they can afford to further their own careers.