Since this test is designed to identify cancer very early, a likely solution to the low PPV problem would be to re-test the individual, perhaps even multiple times. The likelihood of multiple consecutive FP test results, barring a biological reason for the FP, would at some point warrant a more accurate test, be it a screen or diagnostic test depending on tumor type. Also keep in mind that the PPV of the test would increase with a person's age - an 80 year old is at a greater risk of developing cancer than is a 30 year old. This test would also be useful for people with hereditary cancer syndromes.
The cost benefit would be huge - imagine if everyone could identify their cancer 6 months or even years earlier. Mortality would greatly decrease, as would spending on treatments, etc.