Because any complaint filed by the family against the hospital would have been vicious, full of allegations about sending Duncan home the first time, the conditions of his care when re-admitted, etc., and portraying Texas Presbyterian as an incompetent hospital. The press would pick up on those allegations, which could crush the hospital. Even if the hospital could fight back against the family's allegations during the lawsuit itself, it could be too late.
Frankly, regardless of the merits of the case, settling this one early was a no-brainer for the hospital.
Indeed. ANY and ALL past 'mistakes' would likewise be brought front and center in court, in fact, leaked well before being introduced as evidence.
If Duncan had told them he cared for an Ebola Patient in Liberia, he would probably be alive today. The hospital has some culpability but Duncan is the one that is morally culpable.