As a former molecular biologist who happens to be in the middle of a course on the design/synthesis of biomolecular electronics (biological semiconductors, conductors, LEDs, solar etc.), I wonder if the solution isn't as simple as this:
Essentially all biomolecules are synthesized by enzymes. Most are acted upon by enzymes or have some enzymatic activity during their functional life. Quantum criticality could be a useful property to enhance binding and catalysis at enzyme clefts (or other active sites) by enhancing charge/electron transitions in/on a molecule. Criticality may allow transitions and thresholds to be sharper, snappier, more selective.
"Quantum criticality" is just a label we give to a group of mechanisms (and the structures that encourage them) based on some test. I might label the many things that scare my friend's neurotic but otherwise imposing German Shepard as "Fido-phobic". This category might even be scientifically interesting -- if pulling pranks or stealing from my friend were major scientific goals at this point in time. That doesn't mean that squeeze toys that groan, rubber cubes that bounce erratically, and electric toys that "awaken" at random or after a delay share a fundamental property. They simply have properties that have interesting effects toward a certain goal (keeping her dog from interfering in our hijinks)