I agree with Don_Michel, Gotti was too young and unseasoned to be boss. The Chin is sorta like Barzini. He's old school. Although he's in jail also, he's smarter than Gotti. And there is no love loss. The chin tried to kill him in a botched car bomb attempt. The Genovesses are the 2nd largest and true its due to the Gambinos declining to 2nd. Everyone is declining since the commission trials.
I don't think some of the rank and file will respect Jr or Gene like they did John. Someone will challenge either of them, whoever gets the nod from John. John probably won't even name a successor, he's that insecure.
The other families may try to move in on some of their profit centers, with the Genovesses being the most likely.
Whomever becomes boss, the commission or what remains of it may even have to approve it.
Not much can happen though, the feds have everyone so scared and so many informants it would be difficult for any large scale wars to occur.