On a large scale, the wrong results from such tests don't matter if you test people often enough (say, every day). If you get a wrong result one day, it'll probably be corrected in the next day or two with more tests.
If there was a massive government initiative to manufacture such tests and subsidize them so that everyone got them every day this pandemic could be effectively halted in very short order, even without a vaccine.
Almost 6 months ago, the This Week in Virology podcast went in to great detail about this plan, starting at about 6'20" in to episode 640:
https://www.microbe.tv/twiv/twiv-640/
There was also at least one NYT article about it, and it was discussed on HN here:
https://news.ycombinator.com/item?id=23878954
Unsurprisingly the government did not do this, and we had six more months of deaths instead.