You don't have to culture anything (if we did we might as well just give up). We use PCR to amplify a known section of the viral genome. With the oligo production capacity in this country alone we could synthesize enough primers test everyone in the country 10 times in a couple of days. That is not the limiting reagent. The limit is either sample prep or sample collection. Actually running the amplification shouldn't be a bottleneck, there are hundreds of thousands of thermal cyclers across the country. In my more cynical moments I might say the bottleneck is that some companies might want to make money from all the tests ....
I'm trying to learn a bit more about how testing works. Aside from the logistics (getting the samples to the lab, etc), what are the necessary steps for sample prep? Heating to inactivate the virus? Splitting the sample into RNA/DNA/proteins?
Also, where do you draw the numbers of primer production capacity from? IIUC, the only other things that an amplification needs are enzymes (just DNA polymerase for PCR) and "food" (nucleotides), but that I assume is not a bottleneck?
How difficult is it to safely dispose of amplified samples? Can you basically just dump them in the toilet?