Not sure if I'm understanding what you're saying in your first point.
According to the Swedish study
“Moreover, roughly 30 per cent of the blood donors who’d given blood in May 2020 had COVID-19-specific T cells, a figure that’s much higher than previous antibody tests have shown.”
So these are all people who had been in contact with the virus.
https://news.ki.se/immunity-to-covid...sts-have-shown
I suspect that you're right when you say that the majority have not had the virus (although they may have in some of the early hot spots like New York). But I'm open to the possibility that a lot more have had it than official estimates suggest based on antibody tests alone.