I was able to schedule my surgery LONG before I had my therapist fax the letter, but my surgeon and therapist have apparently been sending faxes between each-other over patients for over twenty years, maybe longer. My primary care physician also decided to allow my first testosterone injection the week before my therapist faxed in her letter, but my doctor has been my doctor since I was fifteen, so I highly doubt the latter is at all typical or even common enough not to raise an eyebrow.
(ETA:)
By "schedule surgery", I mean, yes, we were able to set the date for when surgery would take place, but no, they said they absolutely *would not* proceed without the letter. So yeah, one probably wouldn't need a letter to just set a date for surgery, but after the date is set, you'd need the letter to be able to go through with it.