No, this is the wrong argument. What would happen is, eventually people just wouldn't believe any of these tests and they would become useless, even when they did reach a high level of PPV. Also see the point about biopsies - definitely not non-invasive, often require surgery. Also cost - you're now looking at 5-10k extra charges.

False positives can be actively harmful. Unnecessary biopsies lead to unnecessary infections and stress and such.

Cancer is also harmful, particularly if not found early.

I assume every doctor's visit sees you getting checked for every type of condition every time, or do you want your doctor to play the odds and only look for things you're likely to have?

There's plenty of harmful ways to die, and you'll find they don't check for most of them until they have a decent reason.