The economic argument doesn't change whether it's a private cost or purely a social cost (private costs are included in social costs, since private expenditures are part of society's expenditures).

I think the issue is exactly the same no matter who pays.

To reframe it from the individual patient's perspective, when you take a test simply for the sake of screening, there is the chance you'll learn something true that helps you, and the chance you will learn something false that hurts you