Tests cost money. They have blood enough to allow them to discriminate. If they were lacking, they will be less picky: in a catastrophic event they call for blood donors and they rely on tests, risking a detection window.

But under normal conditions, letting only the best candidates donate is the most efficient way.

Heuristics work, why would you not rely on heuristics?

Sounds like something you should evaluate with a cost/benefit analysis, including the false negative and false positive rates.