Didn't the Mafia start off as freedom fighters in Sicily way back? Mafia is a US brand, a well as a Sicilian and Italian one, so that means it has a place in Americas history, which gives it a massive amount of leeway.

Didn't the Mafia start off as freedom fighters in Sicily way back?

To take on your question literally:
While ordinary citizens or organizations have a legal right to privacy, "public figures" or organizations don't. So, for example, if a newspaper or TV station uses your image without your permission, and you can prove that doing so caused you damage, you can sue. But if you're John Gotti Jr. you're a public figure, and the law assumes that, as a public figure, people are interested in you and have the right to see your image without having the media constrained.

Mafia families may think of themselves as a "secret society," but their criminal activites have been exposed so often and for so long that they're in the public domain. It's inconceivble that a Don could bring the news media, or government prosecutors into court and say, "We're a secret society and you're causing us damage by exposing us to cameras, news stories, indictments, etc." It's not inconceivable that a Mafia guy who was exposed as such in a news story, could sue the news medium for "defamation of character," hoping to win damages but mainly to throw the media and government off of the track. But if he did take his case to court, he'd have to prove damage--which would expose him to defense interrogation and investigation. His own boss probably would whack him before he went to trial. And, as dt points out, a Mafia guy's beef with a civilian would be settled with baseball bats and bullets.

Movies and TV series are somewhat different. A documentary is like a news story. But entertainment, which can include fictional dialog and events, is different. Did you notice that in the movie "Goodfellas," Jimmy the Gent Burke is called "Jimmy Conway" and Paul Vario is called "Paulie Cicero"? I'm guessing here but the probable reason is that both were alive when the movie was made. Even though they were public figures and had been convicted of crimes like those shown in the film, the fact that actors played them and had them uttering dialog that they may not have said in real life could have opened the studio to a lawsuit by them or their families. Another possibility: considering that Henry Hill was the source for this material, the studio had good reason to take everything he told them with a barrel of salt.

It's interesting that you ask this question because many legitimate organizations use the law to protect themselves--and mount the equivalent of rackets--in ways that are far more effective than the Mafia's way of doing things:

A business, to be legally incorporated, has to be granted a Certificate of Incorporation by the state it's registered in. Actually, most of the benefit of registering goes to the business, because by legally incorporating, its officers enjoy immunity from certain kinds of lawsuits and even criminal prosecutions. For example, you can't sue an electric utility if the power went out and the food in your refrigerator spoiled. If a meat packer's tainted meat killed consumers, the company could be sued, but its officers or workers could not be prosecuted criminally unless they were proven to have knowingly and actively violated laws. And senior officers routinely avoid taxes by getting lots of compensation in the form of stock options (subject only to 15% capital gains tax when sold, not the topmost income tax rate) or via interest-free loans. Houses of worship and charities get tax relief in return for registering and following certain regulations (such as not contributing to or endorsing political candidates, not using donated funds for pastors' personal gain, etc.). Foundatons can be a real racket. If you form and register a foundation, all contributions to it are tax-deductable for the contributors. The regulations say you can't take money from the foundation for yourself. But you can use your foundation's money to buy a house, register it to the foundation, call it the "headquarters," and live in it. Ditto furniture, appliances, cars, etc. The Mob could take lessons from them.