This is actually answered in my response to your previous question, but it is a good concept that can take some work to wrap one's head around, so I will go in more depth.
Please note that no one other than the individual's physician can advise on what next steps to take once a positive finding is identified - regardless if the positive result was found incidentally or not.
This answer is about the fundamental concepts of screening vs diagnostic testing and incidentaloma management.
Screening vs diagnostic
For the most part, a medical test is a medical test. There are only a few tests that only have application in a screening setting - most are just a test to look for something specific. For example, the PSA blood test just gives the serum concentration of prostate-specific antigen, and can be used for both screening and diagnostic purposes.
A test is defined as being "screening" when it is applied to an asymptomatic individual to search for signs of disease development prior to onset of clinical symptoms, in the hopes of catching disease early.
A test is defined as being "diagnostic" when it is applied (1) in the diagnostic process of a symptomatic individual - or just in general when used for monitoring progress, response to treatment, etc.
In public health and population health, the principles of primary prevention, secondary prevention, and tertiary prevention are related to those definitions.
Screening tests are one of the most important tools to improve the health of a population, and the considerations for whether to apply a test to everyone within that population depends on that benefits vs harms balance previously discussed. The USPSTF, CDC, and academies like ACOG AAFP AAP AHA etc etc review the research and make recommendations on whether a screening test should be given to everyone in the population capable of developing the disease or not. It is not always simple, the agencies don't always agree, and ultimately it is up to the individual and their physician to make an educated decision for that individual's situation. Physicians are trained in these discussions and decision making processes within their scope of care.
I think that what you are referring to is an incidental finding of a mass on a test that was done for another reason. This is commonly referred to as an "incidentaloma" in the medical field, and there are also detailed guidelines on management of incidental masses depending on the type. For example, transvaginal ultrasounds done as a diagnostic test for abnormal uterine bleeding can *incidentally *find an ovarian mass that was previously unknown and asymptomatic. In fact, many transvaginal ultrasounds find incidental cystic masses with a stereotypical appearance that are 100% normal ovarian cysts; however, depending on the features of the mass (or a risk factor of the individual) then additional testing may be indicated.
In that case, the ordering physician (or a specialist they refer the patient to) will go over a risk assessment process - often following an algorithm specific to the mass type - that adds more data to the picture. This process usually includes additional history (like family members with related cancers, smoking etc, job exposures, etc) and/or additional exams and tests (like ROMA), and provides additional information to guide the patient and provider in how much that **INDIVIDUAL is at risk** of the incidental mass actually being cancerous.
Note that some incidental masses won't need additional testing because something about its site/organ/appearance either is (1) so high risk just by existing that they go straight to biopsy or even surgical removal, or (2) so low risk that they rarely/never are harmful so they are just monitored or ignored.
This individual risk assessment is quite different than asymptomatic screening for secondary prevention at a population level. Once a mass is identified, the term "screening" is thrown out the window. Along with the term screening, the concepts of harm/benefit of testing change as well, now that the cat's out of the bag, we've got a positive finding and now we have to decide what is best for this individual based on individual factors.
This concept is exactly why medical professionals usually caution against ordering testing that does not specifically answer a question related to a recommended screening test, presenting clinical symptom, or existing disease. Otherwise, if an abnormal result is found in an asymptomatic otherwise healthy individual, now we're left with the questions: what do we do about this now that we know? What would have happened if we had never noticed it? Is there more potential for bad outcomes if we do more testing, or if we just watch and wait? The answers differ by finding and by individual, so I refer back to my statement:
No one other than the individual's physician can advise on what next steps to take once a positive finding is identified - regardless if the positive result was found incidentally or not.