You may wish to familiarize yourself with the controversy. It is by no means clear that alcoholism is a disease, though there are certain reasons why it is practical to treat it as such – to destigmatize it, to allow insurance payments for treatments and overall to help alcohol abusers over certain psychological barriers to recovery. The problem is that the disease model is nearly a religious doctrine in AA. Members basically subscribe to the theory as a condition for membership, and much of the evidence supporting the theory comes from that self-selecting group. Because of the utility of the disease model in helping recovery, many people are emotionally invested in it, as though questioning it means they can’t possibly recover. The South Park episode in question doesn’t deal with this aspect of it so much as the religious aspect, but the same problem plays out on a religious level.
The episode expressed the belief of the authors that thinking you were “meant to be” something other than what you are is absurd, and it pushes this belief in a reductive, hyperbolic and often disturbing manner. To refer to this as “hate” is to add another level of reduction and hyperbole.
They don’t always present a balanced treatment of a subject, as generous to those they skewer as they were to the Mormons, nor do all the episodes have the subtlety of the “underpants gnome” episode in which the moral conclusion was given by the very people who were collecting underpants on the grounds that it would somehow lead to profits (a useful metaphor for a lot of things, but I read it as a sly dig at the ‘trickle down’ theory of economics). They do not present in transgender episode a good case of someone for whom changing gender would be a positive move. Instead we get Mr. Garrison, who is a worst case example of someone demanding gender reassignment on a neurotic whim. If there had been a more positive case in the episode, would the criticism not stand that easy access to gender reassignment would attract a lot of Mr. Garrisons?