I have indeed been in the situation you describe several times.
If the crew were OK and fit to operate we continued. On 2 occaisions last Summer the crew were not prepared to do so, so we off-loaded and waited for International Rescue
On neither of those 2 occaisions was my decision questioned - even slightly.
I completely agree that rostering days very close to Max FDP limits, particularly in Summer, is an event waiting to happen. So when it happens, the network has to cope with the aftermath!