Let me try to draw an analogous situation.
Suppose a patient visits their GP, and complains of severe, chronic pain. The patient tells their GP that they have lived with it for years, it isn't getting better, and that if they aren't immediately treated with narcotics, they may kill themselves. Should the GP prescribe the drugs?
It should be evaluated on a case by case basis. There are certainly some patients who are in immediate need of narcotics, just as there are some transpatients in immediate need of hormones. There are also, however, some patients who believe that they desperately need these drugs, but are mistaken. It's the doctor's responsibility to prevent the patient with histrionic personality disorder from obtaining hormones just as much as it is their responsibility to keep psychologically addicted patients from receiving opioid drugs. In either case, the prescribing doctor would be culpable for damages incurred by the patient due to their negligence.
Should the 3 month period be a concrete rule? I say no, but I think that it's a guideline that professionals should keep in mind when treating patients seeking hormones. As I said, there are exceptions, and it should remain at the psychologist's discretion to determine when these cases are. Similarly, there are some individuals who should receive a longer evaluation period, just as there are individuals receiving narcotics who should undergo stronger monitoring prior to prescription.
Protecting people from themselves is an important aspect of society, especially in the case of individuals who are potentially mentally ill. In recent years, it seems that we've decided that everyone knows what's best for themselves; that's bull->-bleeped-<-, plain and simple. Just as I wouldn't allow meth and crack et al. to be entirely legalized, I don't think that a patient should be allowed to walk into a doctor's office and demand a particular treatment because they've decided that it will serve them best. These people are professionals who have gone to school for a very long time to learn how to best serve their patients, and claiming to know better than they do because you've read wikipedia, or posted on an online forum is ludicrous.