Public healthcare policy is absolutely not something personal.

It involves costs of healthcare for all people involved, workload on health professionals, hospital occupancy, etc and so forth.

If the rate of false positives in these tests are too high, people that need treatment for their actual illnesses might be on a waiting list because too many are doing follow-up screening and biopsies for non-issues.

And to address your silly "collectivist" fear-mongering, your hyper-individualist mentality is a societal disease. We could do with some more collectivism, in the sense that people have a better understanding of the constraints and conditions of the society they are inserted in.

Would you take a test if doing so statistically increases your probability of death?

Is it moral for a doctor to give a test they think is going to increase someone's chance of death.

It's not personal, it's perfectly rational statistics, i.e. epidemiology. Designing screening strategies is not an amateur's game.