Another thing about widespread testing is then you can see how many people really have it. For instance, in Iceland (and Vo, Italy), they found that 50-75% of the people who have it are asymptomatic but can transmit it (not sure how long that's for, though).
Other countries also used this to isolate the asymptomatic carriers. Basically, if you isolate everyone who has it, you can limit the spread.
Note that you have to test negative people multiple times, though. They can get positive after a while, and the tests have false negatives.