Coronavirus is a virus after all, just like the flu, there is no medication. You can only treat the symptoms, but not the virus itself (as opposed to a bacterial infection where you can attack the bacteria with antibiotics).
So with these undeniable facts it's fairly simple, if your symptoms are so mild that you can cure it out at home without special medical equipment like a ventilator, then you don't need testing. You just have to self isolate yourself for two weeks, get over it and afterwards you are fine and not contagious, so no way of further transmission and therefore no need to test.
If however you develop severe symptopms and need hospitalisation, then they have to test you so they know if they need to isolate you in hospital away from other patiens and protect staff or if you can get a bed amongst everyone else. That is really the only reason.
Your treatment in or outside hospital does not change at all whether you have breathing difficulties because of COVID-19 or any other influenza.
So that is it.