To the best of my knowledge Andronikos' eldest son (Manuel I think) was a decent military leader and lacked his father's paranoia. On the other hand, he didn't seem to be anything special, and the fact that he was killed during the purging of his father's regime shows a potential lack of ability to respond to changes, but as I have said, a corpse would have been a better emperor than Alexios III, so I doubt that that in the 10-20 years left in his natural life that Andronikos' son could destroy the empire nearly as totally as OTL. Looking further ahead, you have Alexios and David, OTL rulers of Trebizond to look forward to, who were very capable and once again not crazy, so by then things are looking up for the empire I believe.
As to why Andronokos went into a paranoid frenzy in the first place, one can only guess, though it could have to do with his own muder of Alexios II. Rather than make Andronikos change his policies, it may be easier for him to just die or be incapacitated by disease than to make him successful, an then hope for the best from his son.