I think you may be confusing what the meaning of a true conflict of interest is, with the practice of nepotism, which happens all the time.
Usually, the only times that conflict of interest laws come into play with regard to nepotism is in a municipal or other governmental jobs or industry under government contract that must adhere to their hiring policies. Even with those, it's only a handful where the practice is legislatively barred. What I mean by "industry under government contract" is when a large corporations hiring for those specific government entities like the energy industry...they will either electively or by legislation must adhere to their anti-nepoitsm rules when hiring for those projects.
Even anti-neposism laws do not typically prevent an individual from helping a friend get their "foot in the door" for an interview when they have a connection to someone who works in the industry; however, in these cases (if an anti-nepotism law is in place) it is really just dictating that their association can't guarantee them the job over a more qualified candidate. It doesn't say that all things being equal, having a connection won't be the deciding factor, because it often will.
What those laws are really meant to do is prevent the use of ones power to trade for professional resources (jobs) or financial or other personal gain in a regulated or government industry, or when a fiduciary duty exists (eg. a chief of staff for a governor telling a corporate entity that if they give their political support to the candidate that the nephew of the CEO will get a judicial appointment when the candidate takes office). That would be a true conflict of interest. And even still, that happens a lot. Most times when a judicial opening happens, the word will get out before anyone can apply (to those in the know) that the job is already promised since in my state these are gubernatorial appointments. It's just an unfortunate fact of life.
The employers/jobs/industries you have used as examples are not subject to legislation nor do they owe a fiduciary duty to anyone that impacts their ability to stick to the "Hollywood elite'' when hiring for those type of jobs. People will inherently be more interested in seeing a famous person's kid than just a new face. In private industry, there is nothing at all wrong with hiring who and what you know. Also, you are referring in your list are talented families. It may be partly that these people's families know the right people to get them in, or at least give them a fighting chance in a highly incestuous and competitive industry. But once they get their "shot" (which is the hardest thing to get) they may be the most talented. It's subjective. It may also have to do with just having been born into those famous families of artists...growing up watching them do their craft...that being their daily "education" needed to gain the skill and knowledge to develop their talent to make them the best choice. Regardless, it's is perfectly legal and if there were ever an industry that nepotism is alive in, it is the motion picture industry (or really any performing arts industry).
Since performing arts is part of your example, i'll note that it is also one of the only industries that can discriminate based solely on sex or race. If a director envisions his lead actor to be a short, black, man, they don't have to consider men of other races, or women, or people over a certain height or of other ethnicities. The job itself is personal characteristic specific. That is part of the artistic process and the law doesn't disallow that. That job description could never exist in any other context.
Most places have little if any jurisdiction to control the general hiring practices of private industry when it comes to who they can hire. Certainly there are laws relating to reasons behind not hiring someone, but that is different. Private employers can mostly do what they want so long as they aren't discriminating based on a protected reasons (race, sexual orientation (in many states), religious or political affiliation, gender).
Look at the highest office in the world, for example - the President of the United States. When a new president gets elected, the entire executive branch and if he gets lucky, some judicial seats too - gets culled out, getting rid of the old team beholden to the past administration, and is then filled with the new executive branch consisting of friends, trusted advisors, political allies (people he owes political favors too) and judges who will uphold the laws they want passed. This is true nepotism at its greatest (or worst-depending on your perspective). I bet that if someone did a chart on the connections to the highest paying and/ or most powerful jobs in government, or private industry to the biggest government contracts, PAC's, and largest donors, almost every big job with be no more than 2 degrees of separation away from the most powerful people in the senate, judicial and executive branches or the biggest campaign donors. Now that would be an interesting study!
Although nepotism is definitely still alive and well, certainly not as as it once was. There was a time when civil service was most known for this, which precipitated many of the anti-nepotism laws that exist in some lower forms of government (like state and municipal) in the first place. There was a time when many an unqualified people would get the civil service jobs for no reason other than having a person "to get them in". This practice bore the saying, "it's not what you know, it's who you know" that matters!. To a big extent. this is still very much the rule.
"More generally, conflicts of interest can be defined as any situation in which an individual or corporation (either private or governmental) is in a position to exploit a professional or official capacity in some way for their personal or corporate benefit." Just as a note, the legal definitions pertaining to conflict of interest law are nuanced and complex. IMHO, I think Wikipedia is OK to get information that will help get a starting point for research, but never as a main source. Especially when the abiding authority is just as accessible, free and online (like a legal dictionary or treatise).
Helping to get a friend or family member hired into to a sought-after position is generally not a conflict of interest. The "duty of loyalty" you are referring to is when a fiduciary or professional duty exists and someone does something that can adversely affect the person or corporation they are supposed to be protecting, expressly for their own personal financial gain. While I can see how you could think that this (you example) could fit the bill, from a legal perspective it really doesn't.
Some professional organizations disallow even the appearance of impropriety or conflicts (like lawyers, doctors, stock traders, etc).