Why would a doctor pay another service to order labs for them?
Sadly there are a lot of clinics popping up to serve the internet self-diagnosers, but not in the way you're talking about. They're built around a single enterprising doctor who either believes the disease is undetectable by normal tests, or who is aware that they're stringing patients along but likes the money. If someone opens a specialty clinic for an internet-popular disorder, they have a perpetual line of patients who will gladly pay for a doctor to tell them what they want to hear.
Whats that supposed to mean? Most countries have private healthcare too. Sometimes it is as popular as public i.e. Australia 45% is private vs 55% in US.
it's an attempt to starve funding from the public system. (Howard was responsible for destroying the first universal healthcare system in australia before the later second attempt)
The private hospitals specialise in elective uncomplicated day ops because the returns from the medicare payment are better than complicated cases that get the same fixed payment. It's also why they transfer anything slightly complicated into public hospitals.