Would you bet your life that reputation alone was enough to keep an airliner from crashing? We know right now airlines shave every possible extra cost to achieve the lowest prices because passengers are very price sensitive.
Even more, right now a lot of people don't know they're flying on any number of regional airlines, some of which may have terrible reputations when they fly "United express" [1]. Consider that one of those airlines beating a passenger to get them off the plane didn't change the situation.
Of course, now we have safety standards rather than just reputation preventing crashes, which seems like the appropriate thing.
Please review the history of the DeHavilland Comet and the Lockheed Electra for what happened when an airliner acquired a reputation for being unsafe.
Would you fly on an airplane that had a reputation for crashing? Me neither. Crashing airplanes is very, very bad for profits.