Obviously there will be false positives and false negatives with this app. If you could catch even 50% of cases though it would go a long ways towards stopping spread. The question is whether you can get that without a huge false positive rate. My guess is that you can, since it appears that the primary transmission mechanism is sustained indoor contact, within about 15 feet. You could probably increase the accuracy if you could detect being indoors.

On the cost issue, the obvious solution is to make testing free and easy, and to provide a financial incentive to get tested quickly after the app alerts them.