You say that despite all of the evidence to the contrary over decades in this country and elsewhere. Why do you use the phrase "professional doctors"? You you think professionals are less prone to human frailties that the rest of us plebs? Take a look at the tax defaulters list and, this may shock you, but doctors pop up with great regularity. Work on the assumption that they are just the same as the rest of us, no better or worse. Very few are ever sanctioned, particularly is specialist areas.
Diagnosis is based on their opinion after looking at multiple factors. They can just play the emotive card, they are good at that, and talk about the unfortunate people to whom they are peddling their wears rather than the large incomes they are getting from their customers. You will never be more vulnerable, more open to grasping at straws, than when you sit across from them and they are offering you the chance to have a family.
While you may be slightly more reticent when it is your money there is no chance of rationality coming into it when they are asking you to gamble with someone else's money.
If it is run by a public only system with everyone concerned getting a flat salary with limits on the numbers of times each person/couple can access the service and with a limit on the overall number of IVF rounds the State funds each year then maybe it could be done but the most likely outcome from this is rich doctors getting richer.