I think that the whole talk about herd immunity would probably be a good reason to test the entire population (i.e. everyone getting a test at some point). I think that would gauge just how many people got the virus at some point. Although there definitely isn’t enough capacity to do this at the moment, maybe this would be worth doing when sufficient 20 minute tests become available, something that would be very useful at Schools and offices.
The current tests (and AFAIK the spit tests) don't test for if you've had the virus, only if you have the virus at the time of the test (although in some cases it is picking up dead viral material from cleared infections). All that testing the entire population does is tell you who has the virus during that week you've tested everyone
You can't even test for immunity with any degree of reliability as antibody levels fall the longer it is after the infection, and there isn't a T-Cell test.