Yes, it is a real problem. Cancer treatment is generally not a walk in the park, and you'd be administering it to people who never would have otherwise need it. Imagine having portions of your body removed and/or receiving brutal treatment for no real purpose. Further, these tests have false positives and even in the best case that could cause undue stress and strain.
It's been a while since I've been in the field, but at least as of several years ago the oncologists I knew were starting to turn against heavy pushes for early screening for these and similar reasons.
My understanding is most people get cancer cells regularly. Possibly every day. The immune system clears them out naturally. The question is how old is a cancer before it becomes a threat worthy of analysis. Obviously one cell is probably not interesting. I think there's evidence even cancers that are large enough to be visible to the naked eye are not typically dangerous. If 99% are harmless, what is the point in early detection? It's an expensive test and expensive doctors visits to mitigate a 1% risk. It's interesting from a research perspective to be sure, but not as a standard of care.
We don't generally treat neoplasms, because we get lots of them, most are of no consequence, and treatments are damaging. We care about the malignant neoplasms where the biological cost of treatment is outweighed by the cost of allowing the malignancy to remain.
If you can reliably distinguish between those malignant neoplasms and "any growing neoplasm in the body", collect your Nobel and enjoy your well-earned message board karma from winning this debate. Since you can't, what you're really doing here is begging the question.
It's not about superstition or fear, it's about resource allocation and common sense. Medical services are finite. Screening and continuous monitoring of every potential tumor in everyone would be crazy time consuming and incredibly wasteful. So let's not.
EDIT: Just to be clear, the idea of accurate early detection is of course a good one. But as of today it is just not viable.
You also have to be careful to quantify "useless," "useless" can mean "doesn't work," but if we are objectively detecting cancers it's hard to say it's "useless."
You also may have one person's life saved for every 20 who were harmed by the screening program - it's still useful for those people who benefited. So yeah, that terminology is problematic.
It's more accurately to speak of a bad screening program "causes more harm than good on a population wide scale."
And, of course, it's important to clarify we are talking about screening the general population who is asymptomatic and at average risk. If you've got symptoms then the topic becomes diagnosis, not screening. If you have an increased risk due to genetic factors, family history, or increased exposure to some cancerogenics then then risks of screening for the general population don't apply to you.
Then why not respond appropriately, and stay aware of cancers but don't treat them until they reach some critical stage? I don't see how intentionally remaining ignorant is helpful.
Because if a localised cancer is destined reach later stages then getting treatment as soon as possible is of the essence - if you wait for it to become malignant before treating it the chances of survival go down.
Often it's impossible to tell right now which asymptotic localised cancers are going to eventually kill the patient if left untreated and which are going to be non-bothersome or even spontaneously regress. They look the same in many cases.
What would you do if your doctor found a cancer and refused to treat it? What do you think most people would do? I reckon most people would shop around until they found a doctor that would treat it. Even if they followed the doctor’s advice, the knowledge they have cancer might be incredibly stressful and distressing.
On a population-wide scale, ignorance may be the best option right now.
If my doctor said "Yes, this is cancer, but it's so small and more likely than not to never become a problem. Come back in two months and we'll check again to make sure it's not becoming a problem", then I would be fine with that answer.