I think in a controlled situation, such as a dog in a rescue, it could be rehomed. Simply because biting the mailman is like this--
The dog sees mailman everyday, kids say "Go get 'em!", laugh, And otherwise rile him up to the point where he's super, SUPER frustrated because somebody is constantly invading his space, and there's nothing he can do about it. Until the day he gets out, that is. And the family is teaching him to be the aggressor.
I'll bet, that with someone experienced, the dog could be broken of the bad habit in a few days. The dog never would have started in the first place. However, because of this family's stupidity, the dog has now "Bitten." and become a dangerous dog, incapable of being rehomed.
But I think it's irresponsible for a family like this to rehome a dog like that.