Gotti was pretty sharp

The consensus these days is that the only thing sharp about Gotti was his clothes. He liked publicity and liked to hear himself talk and be talked about, something a Cosa Nostra boss definitely should not do: this kind of behaviour didn't end well for Al Capone and Joe Colombo, either.

wherever he held court

Which was the Ravenite club, where he actually avoided saying anything compromising. Important talks were held in the flat above the club (where the widow of a Gambino soldier lived); for some reason Gotti thought that flat would not be bugged. It was Sammy Gravano who insisted on holding confidential meetings in the street. But the thing about "holding court" at the Ravenite was another one of Gotti's stupidities: all Gambino caporegimes had to appear once a week at the Ravenite; all law enforcement had to do was put video surveillance on the Ravenite (some of this footage can be watched on Youtube), and they knew who were the captains or acting captains in the Gambino Family.

more fair boss

Actually, one of the reasons he was killed was his greediness.

who got killed near his limo either by Gotti or his men

Gotti was not stupid enough to kill his own boss himself; but the actual shooters were from his crew, while Gotti and Gravano supervised the whole affair from a car close by.

some guy with a cigar still in his mouth, and the most classic one of all, front page photo of the snitch, lying dead with a dead canary in his mouth.

What you seem to remember here is not the shooting of Castellano and Bilotti, but the killing of de facto Bonanno boss Lilo Galante in the late 70s. He was a really nasty guy. Apparently he had once been a close friend of Joe Bonanno himself (and his underboss), but Joe never mentioned him in his autobiography because Galante's undeniable involvement in the drug trade ran counter to one of the myths Bonanno tried to spread: that the Italian mob was had a rule against drug dealing — a definite falsehood.

I don't remember the CP element

It's in Murder Machine. Probably based on what Dominick Montiglio told the authors, which means this should be treated with caution. Montiglio is known to have lied about other things (his army career in Vietnam, for example); Roy Demeo's son Albert has pointed out that some of the details in Murder Machine are definitely false (although he admits that in general the story is true).