As long as we are fairly confident about the specificity and sensitivity of the test, then it DOES tell us something, even if those numbers are as low as 85%.

Here's the flaw in your proof - If the test has 85% specificity, then the chance of 200 million positives is essentially 0% (you'd get ~150 million if no one really has the disease). Getting a result of 200 million positive means either you DO have a significant number of true positives (50 million), or your 85% number for specificity was wrong.