Yes, one way of understanding E is by considering pain a problem. After all, the phenomenon of medicine itself is likely attributed to the presence of pain. As medicine at the very least reduces pain by potentially eradicating it and its potential additional harm if not treated, we could assume that medicine is a solution to health caused by pain management. As pain is associated with illness, sometimes those of which are life threatening, people experiencing previously hard to treat terminal illnesses benefit from modern medicine that is now able to treat those illnesses. This resolves the problem of not just current pain from an illness, but a pain that could bring on future illness.

The use of “they” in this context is grammatically correct as expressed by the APA and MW Dictionary. https://apastyle.apa.org/style-grammar-guidelines/grammar/singular-they