Not that I have any knowledge of whether anyone is running PCR tests on valid random population samples, but I suspect the concern is that this isn't happening, and that any followup study you could do on these positives will have biases that are hard to correct for?

That won't work. There is a high false negative rate with those tests, and it's likely that false negatives are more common in those who have only mild infections. A study of patients with only positive tests would be a skewed sample.