The direct answer: your dermatologist is giving you the standard advice for your situation, and the four clinics are answering a different question from the one you asked.
They were asked "is this scalp operable", and it probably is. She was asked "is surgery the right move for this person now", which takes in the trajectory of the loss and the sixty years of scalp you have left. Caution below about 25, and in fast-moving loss a little beyond that, is conventional precisely because two things are still unknown: where your pattern will end up, and how quickly. Surgery treats the pattern you have today. It does nothing to the underlying process, so the untransplanted hair keeps thinning on its own schedule.
The conditions that make the wait worth it, named specifically. First, a rapid change over eighteen months, as you describe, means nobody can yet tell where the stable pattern sits, and a hairline designed against a moving target is how people end up with an island of transplanted hair and a gap behind it, which is exactly what norwood3ben describes. Second, involvement of the crown alongside the front matters, because the crown is a large area with high graft demand and a lifetime donor supply commonly cited at only about 6,000 to 8,000 grafts. Spending a third of that at 26 is a decision that constrains every later one. Third, and this is the one people skip, medicine changes the picture: finasteride lowers DHT by roughly 70%, and in the five year data around 90% of men kept their regrowth or had no further visible loss, while minoxidil needs a good twelve months before you can judge it at all. A year or two on treatment tells you and your surgeon something you cannot know any other way, which is whether the loss can be held.
The exception, so I'm not being absolute: a stable, purely frontal recession in someone already on treatment, with an honest conservative design, is operable in the twenties and is done all the time. That is not what you have described. The reading that covers this properly is am I a candidate for a hair transplant, and the medicine side is in hair transplant versus medication.
The discount for booking in July is not a clinical argument, and I would set it aside entirely when you make this decision.