Not exactly. The FAA often doesn't know about it, as it takes that "preponderance" you mentioned. If they don't have that, they "don't" know about it. It only matters what they have relative to that, otherwise it could be harassing operators, which is not ok. If someone makes a complaint, as you say, it has to be investigated. Unfortunately, many people think the FAA sits there with other government agencies and traces phone calls to retaliate against people who file complaints (which can also be done anonymously). What they want is safe operators and safe pilots that follow the rules. They don't want someone to crash or harm anyone due to fatigue or falsifying records/rest. All to often, pilots accept risk and missions that violate the regulations (in the OP example), or they do something they aren't sure about/weren't planning on and don't declare an emergency, then go back and repeat the process until something bad happens or that one time where it doesn't quite work as it did in the past.