First, because it comes off as too blithe and superficial for serious essays. It sounds like something a teenager would throw at another.
Second, because its use borders on hypocrisy. When you apply a pejorative to someone's else's grievances, you are complaining about complaining, more or less.
Third, because it obscures meaning. A charge of "whining!" could be an indictment of style or substance, but many authors who use it feel no apparent need to clarify. If someone has a legitimate grievance, but gets their message across in a maudlin or repetitive tone, it is fine to express your opinion on that matter if you consider your appropriate role to be that of editor. If not, it may be better to focus on content.
Fourth - the subtlest and most important reason - it is easiest to use to unfairly slur those communication styles that are most healthy, not those that are the most obnoxious. I have noticed in recent years a distinct tendency to get one's grievances across through put-downs (aggressiveness) or through intentional distraction put into question form (a form of passive aggression). Whether or not these strategies have been developed to avoid the ubiquitous charge of "whining", I don't know; the point is that both aggression and passive aggression are unhealthy.
Consider the following (completely fictitious) scenario. A man has just come from a 12-hour shift on a construction site. In the past, he has offered to lend his lawnmower to a roommate, noting that it is quite dirty, and stating vaguely that "You or I could clean it; shouldn't be too hard." Neither remembers the conversation verbatim.
Roommate: Hey, your lawnmower's totally filthy, dude. I need to use it. Hop to it, dude.
Man: Oh, you can clean it if you need it right now. I had a long shift and I'm not inclined to do it this evening.
Roommate: What? It's your lawnmower. Why can't you clean it?
Man (version 1): You must either be deaf or a moron. I said you can clean it.
Man (version 2): Why should I clean it?
Man (version 3): I'm too tired to clean it right now. I need a shower.
Ask yourself which of these versions is the healthiest (i.e., most assertive). Now ask yourself which of these versions is most vulnerable to the charge of "whining".
Version 1 replies to Roommate's fallacious complex question with insults and repetition. It doesn't move the conversation forward, but it can hardly be accused of "whining".
Version 2 is just a "question", albeit a fake question (there is no reason the Man should clean it, so it makes no sense to ask for a reason). Most modern people were brought up in highly ideological institutional schools where it was constantly drilled as a mantra that there is "no such thing as a stupid question", and so they become habituated to putting most of their jabs and ripostes in question form as a form of rhetorical defense.
Version 3 is the most informative and least obnoxious, but in the modern world it would surely invite the knee-jerk reply. It is also the sort of reply I hardly ever hear any more (outside of my marriage and certainly highly-scripted "professional" discussions). To constantly scream "Whiner!" discourages people from being assertive or even minimally informative, and encourages the verbal war of all-against-all that seems to have replaced discourse.