Coming from my friend who works in a hospital on why they aren’t testing.
First, if you say you test you have hundreds of people who show up for testing with no symptoms. It’s allergy season, yes you have a sore throat, your eyes water just like every other year. Go home
next they only have a limited number of tests, which they are using to confirm people who are critical and need admission. If you have a mild case, stay home and don’t infect others
also, what is that test going to tell you? That you’re sick, guess what you already know that. There’s no treatment, no medicine, so self isolate and try not to die, or infect others.
Also, if one member of your household has it, chances are incredibly high that everyone else in your family does too. So go home.