This isn't a trivial problem. These patients are often asymptomatic, and objective physical exam findings are not easy to elicit or entirely non-existent. At this point, the only thing a doctor can do is screen every patient coming through the door for cancer via CT, MR, mammography, colonoscopy, etc., but this doesn't translate into clear cut clinically meaningful benefits that helps patients live longer. You often end up irradiating non-cancer patients and creating new cancers, or you selectively diagnose non-aggressive cancers (length-time bias) and unnecessarily put patients through surgery, chemotherapy, and radiation (i.e. over-treatment). Some fraction of these patients then go on to live shorter lives as a result of the over-treatment. This particular issue we're discussing is more of a medical science problem than a healthcare system problem.
Anecdote from med school: on my surgery rotation, I had a ~70 year-old patient recently diagnosed with early stage colon cancer on a screening colonoscopy. He had undergone chemotherapy and was now about to have the cancerous part of his colon removed. Surgery was a success, and the patient recovered in the hospital for a few days before we sent him home. The day after he goes home, we found out he died from a massive pulmonary embolism, likely a post-operative complication from surgery.
Alternatively, had this patient not had a screening colonoscopy, we never would have found his early stage cancer. Based on his cancer stage, he likely would've lived another 10–15 years without treatment (age ~85-90). Instead, our interventions killed him a few months after diagnosis.