Even though I'm one of its defenders, there's MIND OVER MAYHEM.
In spite of all the blunders Marshall Cahill makes, what makes him confess is Columbo arresting his son on a trumped-up charge (which he later ADMITS is a trumped-up charge), so why WOULD'T Marshall confess? (A little like Ned Diamond confessing to cover for Grace Wheeler, except that it's covering for someone who's actually innocent.) And according to Columbo, he probably couldn't get Marshall any other way, as in evidence.