What appears to be the potential danger of a high false positive is the narrative that it's safe to end stay-at-home because everyone already has/had it. Which seems to be the story being pushed with any talk of a lot of positives.
Implementation examples: South Dakota and Sweden.
It seems clear that the infection rate has been severely under counted, meaning mortality rates are artificially high. Even better (from the standpoint of restarting) that "flu-like" mortality rate is concentrated in people over 60 years old.
What that all means together is the economy can easily restart, with the most vulnerable (old and sick) taking extra precautions.
This entire thing has been a fascinating exercise in how poorly central planning can work given bad information. The cure has been vastly more damaging than the disease.