If you want to contact trace, we would need many orders of magnitude more testing.
If you want to know prevalence over time, and therefore Rt, we could ascertain it very accurately with much less testing, but it would need to be done against statistically valid and unbiased random samples of the population, not at all what is going on today.
Since the testing we are doing today doesn't really satisfy either condition, and there's also no front-line approved treatments for those that are definitively diagnosed (rather, we treat the symptoms, which are self-evident), frankly I'm not sure I understand the extreme focus on testing.