One other thing to note with the antibody tests is that there appears to be some cross-immunity between the novel coronavirus and established, endemic coronaviruses. (Whether having antibodies from a seasonal coronavirus gives you meaningful protection against covid is another question, one we don't really know, but we also don't even know if antibodies from covid give you protection against covid.)
If a serological test is insufficiently specific, you could just be picking up positives from antibodies based on the cold that HCoV-OC43 or HCoV-HKU1 gave you back in December and not antibodies from exposure to SARS-CoV-2. The test kits being used in these studies were rolled out quickly and were never approved by China's FDA, the only thing we know is they're not just a box of rocks (the Stanford study had 30/30 positive as a validation for people who previously tested positive for coronavirus viral load, but that doesn't necessarily mean it's not testing something else either; I could give 100 people who had coronavirus a chicken pox serological test and it'd come back 100/100 too, which says nothing about how effective it is at testing coronavirus).