There are enough people who were PCR positive that antibody tests aren't necessary for this kind of study.

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.

Widespread antibody testing will not be sufficient to answer that question. It appears that a significant fraction of patients who recover from asymptomatic SARS-CoV-2 infections don't produce detectable levels of antibodies.

You may also have some level of antibodies without actually getting sick like what happens with tb. Besides if vaccine is eventually developed antibody test will be useless

Source?

Thanks for providing a link. For anyone else who comes along, the supporting statement isn't very prominent; you can find it in:

"If you test negative" > "You could still have a current infection." > 2nd sub-bullet