Exposing asymptomatic potential issues leads to medical care that often does not meet out standards for medical tradeoffs. Chemo is nasty, even the most minor surgery has risks. We endure the risks because we are addressing either major health issues or other dire uncertainties. Using our heavy duty treatments for issues without any symptoms at all would, normally, cause the patient suffering in excess of what would be justified. Chemo is a life saver when it's saving lives -- if the alternative is no symptoms, it just ruins your life for a profoundly uncertain upside.
Well, you've reduced a rational position to absurdity here. The concern isn't that someone is going to require chemotherapy. The concern is that an asymptomatic condition will go undisclosed until it is symptomatic, at which point preventative treatments are futile.
Even if we accepted your example, there is a fundamental inconsistency, since doctors in America regularly prescribe very serious drug interventions for patients who essentially self-diagnose for being neurotic/adhd/etc.--actually a much worse problem than the one you're so eager to curtail.
There exist both type I and type II errors, yes. But it's not absurdity, the South Korean Thyroid Cancer "epidemic" is one of the classic examples of the dangers of using traditional treatments on the lower-severity cases more advanced scanning uncovers. It was a public health disaster with lifelong consequences for those impacted, and was, legitimately, people getting cancer treatment for small cancers so harmless that they would have died of something else before becoming symptomatic.
If you seriously think this is a realistic scenario, you should call your doctor and tell him you've decided you have cancer and would like to order chemotherapy. See what he says.
Nobody is deciding for themselves to do chemo because they think something looks funny on a scan. All the scan gives them is a reason to talk to their doctor, who will do all the usual due diligence before deciding on treatment, if any.
And on the flip side, I hear stories all the time of people who DO have symptoms but they get dismissed by their doctor as stress or food allergies or whatever until it's too late. Maybe if patients were armed with a scan that shows a mass at the site of their abdominal pain, there would be fewer of these horror stories.