And no one benefits from faulty logic.
Not so!
Faulty logic didn't come about because people were stupid or incapable of 'explaining things' through rationality but because they benefit from 'explaining things away' through rationalization.
If faulty logic were not beneficial, then it simply would not occur, with the exception of conclusions drawn by morons, or even by intelligent people in cases of extreme difficulty or complexity that require careful reasoning. But as you yourself have said, faulty reasoning does not occur only in these circumstances, but frequently - in the OP's argument, even more frequently than sound reasoning. Faulty logic
is beneficial because such a judgement is beyond the province of pure logic - or more clearly stated - all things percieved by one to be beneficial
are beneficial to him. If one has the desire to lose a leg, removing it is his gain, though others would consider it a loss. That is why judgements of perception cannot be decided through reason.
In light of this, perhaps the first stage of the grieving process - denial - is a positive one, in spite of its blatant irrationality, as a transition period between a traumatic incident and its acceptance may be necessary for the victim's psychic well-being.
Psychic well-being must be considered above rationality itself, because one only applies reason soundly if he feels the most objective interpretation of phenomena is best for his well being. For instance, an INTP making decisions on fanciful whims would certainly become distressed, just as an ENFJ making interpersonal decisions based on rationality alone would be similarly distressed. There is no standard that ranks rationality higher than irrationality, as an argument to prove otherwise would be based purely upon logic itself and would therefore be circular. Paradoxically, a proof for the superiority of irrationality over rationality would rely solely on that wish it attempts to discredit. However, an argument - not a proof - for the same would not be required to be rational at all. For all of the above reasons, unsound reason is every bit as beneficial as sound reason.