There's plenty of room between "go under the knife" and "ignore altogether."

if someone told you, you had a .01% chance of getting a disease for example, aren't you better off with that information? even if it is noisy?

Any test that is approved for use would have a better-than-random outcome distribution. Preconditioned on that, a test result is still useful no matter how uncertain. It is never the case that more information leaves you in a worse position.

I have personal experience here:

Every year I have elected to have ultrasounds done of my major internal organs. In the past two visits, the technicians spotted multiple developing growths in my liver and now kidneys. These are very likely to be benign cysts, but one piece of blood work that could be a marker for cancer is inconclusive. The odds are still high that this is totally benign and will either clear up on its own, or at least stop growing and cause no further issue. Still:

1. I'm getting my blood work done now far more frequently (twice per year instead of every other year), with specialized/not typically ordered screening tests;

2. I am redoing the ultrasound every year to track progress; and

3. I am actually taking advice about losing weight and exercise far more seriously than I otherwise would, as these issues often resolve with weight loss.

I am actually healthier now than I was two years ago, and feel better about my physical and mental well-being. All while staying on top of what could have be a life threatening issue if left untreated and ignored.

I look forward to the day when I can go get a monthly MRI-like scan. That would be wonderful.

No its not. This is extremely paternalistic. Humans know how to understand noise and statistics. You don't get to decide that for me. I want more lives saved with more information.