> And a lot of countries do the test twice, and only act on double positives or double negatives.
The false-positivity or false-negativity of a test is not independent across multiple runs. Its not like each test rolls a dice and decides if it wants to be accurate or not. Its more accurately based on the human being tested; a false-positive test for one human would increase the likelihood for a subsequent false-positive test on that same human. Double-testing helps, but its not the solution.
> How can this be OK? People will spread it if they think they are not sick.
First, do no harm. This often means the first response for doctors is to do nothing (and gather information), until they're certain enough that their actions will not result in harm. Doing nothing is not causing harm; the harm has already come to their patient. The doctor is not the cause of that harm; the world caused it. Its a core responsibility of a doctor to not make it worse accidentally.
Sure, people may spread it if they think they're not sick. But what if a doctor provides a false-negative result to an asymptomatic patient? Now, that patient has been told by a doctor "you're fine". They go back to work. Now, they're spreading it. "My doctor said I'm fine, this little cough is probably just a cold." Patient doesn't go back to get it checked out again.
What if they provide a false-positive result to an asymptomatic patient? "Wow, that's crazy. I must have just gotten lucky" the patient says. Two weeks later, they leave quarantine after getting an accurate negative result. A month later, they actually contract the disease. "But, that's impossible. I already had it! My other doctor told me so, he even gave me a Test." Can coronavirus re-infect patients? Now, we're not so sure. Panic. Research studies. The truth is obscure.
This is how doctors everywhere operate, and this is true at the FDA/CDC more than nearly every other country. Our standards for drug development, testing kit development, etc are among the most stringent in the world. It would take a disease far, far more deadly than the coronavirus to compromise them.