"but my main problem is that I'm having trouble seeing "why" we are learning what we're learning and how it fits together. Why is understanding chirality relevant to biology, for example?"
Check out Thalidomide wikipedia page. The drug was released as an racemic mixture when one of the enatiomers was teratogenic. I'd hope my doctor would know the difference, at least on a conceptual level.
It's the difference between know "mitochondria is the powerhouse of the cell" to "oxidative phosphorylation". Or when acetylcholine is released how it is broken down at the synaptic cleft. DNA base pair binding and thiamine dimers. What vitamins actually are and what they do biochemically. Why are saturated fats bad vs unsaturated, why is one a solid and the other a liquid. Why do steroids pass through and hormones need carriers...
Organic chemistry should be giving you the tools to why biology works and understand problems when it doesn't work. It underpins everything biological. For me organic chemistry was when everything made sense, when you understand things at the most fundamental level. Biology became a lot less handwaiving and representing diagrams as "blobs" instead of what they actually are. It brings you from the elementary school picture of biology to the reality of everything. Gah, I could go on and on. Then if you get into science, how people cure a disease (as opposed to prescribing a drug) or understand a problem, organic chemistry is there.
That said, most doctors don't need it. It's more like "see set of symptoms, prescribe this or that drug".