One of the things I've learned over the years is that an analogy seldom clarifies things. Most of the time, an analogy only serves to make your point more obscure. Often, people reach for analogies when they are unable to clearly articulate their point. The problem with this approach is it only works if the person draws the exact same comparison you are trying to draw but can't articulate. That doesn't happen very often. Generally what you find is that the analogy which seemed so straightfoward to you, means something entirely different to someone else.
Sometimes, the whole point of an analogy is to engage in a logical fallacy (fallacies of composition, usually). Generally these are of the form, "If you except that A is true, and that A relates to be according to some relationship, then you have to accept that D is true, because C and D have the exact same relationship." Of course, very rarely do C and D actually have the exact same relationship and now your argument has all sorts of points of failure that you didn't have before you introduced your analogy.
When analogies are appealed to, typically I find that the resulting conversation tends to be an argument about the thing in the analogy and the correctness of the terms used in the analogy, rather than being a discussion about the thing which was presumably supposed to be under discussion. Analogies are a great way of derailing your point, and I speak from personal experience there.
In this particular case, I don't agree with even the initial assertion that players of gamers frequently appeal to movies as a way of explaining RPGs. I don't even agree that in the statement 'the players in a game are like actors, and the characters are the roles they play' is a particularly common assertion, or that movies are necessarily what is in mind when it is made, or that the analogy - because it attempts to explain the difference between a player and a character - has any applicability when extended to trying to explain anything else about RPGs. Just because 'the players in a game are like actors, and the characters are the roles they play' might be suitable analogy for one narrow purpose, doesn't mean that any other analogy that extends from that to compare movies to RPGs is a suitable one. So I don't have to agree that GMs are either like directors or like producers. They don't have to be much like either one.
Moreover, I don't agree that he accurately defines directors or that its tempting to compare DMs to directors. First of all, what he describes as the directors role is not one which all directors would defend. Some directors believe that is there role, but many directors would say that that is only one approach to directing and that they feel they get better results by not exerting a large measure of control over the lead actors or the editing process but instead rely on the actors to interpret the role freely. In other words, not all directors direct from an authoritarian stance. So in fact, all he's really saying is that not all DMs are like all directors.
You know an analogy is at the breaking point when the person using the analogy realizes how limited of a thing he's actually saying, and then has to dismiss a whole category of objections by saying, "but those are bad examples".
What gets me is the percentage of people who agree with the person's underlying point who, when they read an analogy, seem to think, "Booyah! Take that you people who disagree with me! Now I'm armed with an analogy, and hense now you'll be unable to do anything but agree with me, because really, who could disagree with this analogy!" And then the whole analogy game turns into this sort of logical fallacy - "You like apple pie don't you? Well, anyone who likes apple pie must also like ice cream. And he doesn't like ice cream. Apple-pie hater!"
I think that humans have this natural emotional tendency to act as if the number of analogies you have on your side in some way determined how correct you were. Or at least, I know I used to act as if that was true, and I observe similar behavior to what I used to engage in in others.