And although it would turn up a lot of false positives, it would also catch a lot of cancers very, very early, making survival rates higher and costs lower.
It's not something people like to talk about, but early detection doesn't correspond to improved survival (see, for example, Screening for prostate cancer: systematic review and meta-analysis of randomised controlled trials ).
Forty years ago, prostate cancer was considered a fast-moving, highly lethal cancer. PSA screening was introduced, with a heavy push for annual screening of men over the age of 50. Large number of men tested positive, and a metric shitload of tumors were found and treated. And the death rate didn't go down.
For the past several decades, there's been a heavy push for frequent breast examinations and mammograms for women, the idea being that if breast cancer is detected in the early stages, it's easy to treat and many lives will be saved. The detection rate has certainly gone up, as has the number of tumors treated. The death rate? Hardly budged.
It's something of an article of faith among anti-cancer activists that screening and early treatment save lives. In practice, the vast majority of improvement in cancer survival has come from improved treatments, not improved detection. Most people with early-stage cancer either have something so slow-growing that it can be safely treated at a later date (or not at all), or something so fast-growing that they'll die despite treatment. The percent of cancers where early treatment will improve the outcome is believed to be in the low single digits.