The International Committee of the Red Cross (ICRC) first arrived in Rwanda in 1990. The ICRC's work in Rwanda was primarily humanitarian (food, caring for orphans, etc). In early 1994, fearing that the peace process would fail, the ICRC began to stockpile water and medicine. In Switzerland, a medical team was ready to be sent to Kigali if needed. During the genocide, the ICRC was responsible for saving thousands of lives. Since the genocide, the ICRC has worked to heal wounded victims of the genocide, reunite family members, maintain standards at detainee prisons and generally help repair the country.
In terms of relevance to The Overwhelming, what I've drawn from the information is that, because the ICRC was primarily in Kigali to provide humanitarian services, the Rwandan hospital would be responsible for general medical concerns.