Because with a false positive antibody test then you can still contract and be a carrier. This is a problem for the immunity passport proposals (which I think are a terrible idea anyway, but that’s beside the point)
It doesn't have to be perfect. If 90% of the at risk population where it's allowed to spread has immunity, then that's high enough that the virus would logarithmically decay. That's often referred to as "herd immunity". So long as on average cases result in less than one additional case you're fine. 1000 cases with an uninhibited net reproductive rate of 4 with a population that's 90% immune would result in the next "generation" having 400, then 160, then 64, then 26.... You get the point.
This is especially bad if there's no government-sanctioned method for infection - then you end up with a mess of asymptomatic people trying to get infected while they're already infectious.
