As your last statement here illustrates, this is because of the way most health care is paid for. In a system where third parties pay for most health care, the doctor has to answer to the people who pay even though they are not the patient. Even if he personally is convinced that a particular experimental treatment is low-risk and has some chance of success, he still has to convince the people who pay, which is a much higher bar.
If a person is undergoing cancer treatment their vitals are being monitored anyway. I'm not talking about the person asking for any additional lab tests. I'm just talking about them deciding to fast.
There are lots of low tech/no tech ways to monitor your health and people pursuing alternative remedies routinely pay out of pocket as both a cheap alternative to paying a doctor and because they often don't want a doctor involved in their health choices.
But I wasn't just referring to fasting. It's incredibly difficult to do anything that isn't approved by a doctor. Even lab tests is a huge battle unless you pay out of pocket via mail order.