I've heard enough about, and experienced, the decisions of some of the otherwise well-regarded private ObGyns that I am somewhat skeptical of their motivations. If we were in the States, we'd probably be suing our very popular ObGyn for negligence. As it is, I don't want to put my wife through the emotional distress of going down this route.
I and other parents have had sufficiently shit staffing experiences at top tier hospitals (again, excepting Matilda) that we now regard the process as a complete crapshoot. If you are lucky, everything is smooth sailing. If you are unlucky, then the process sucks. Paying more does not seem to improve your luck. For ourselves, we have experienced both public and private and had better staffing experiences in the former.
Were we to repeat the experience, we would either go Matilda or public. And we have insurance that would cover the former. For myself (and it would not be my decision) it would be partially a matter of principle- we could afford it, but I am sufficiently cynical of the Westerner-oriented medical system that I don't want to support it financially.