Nobody does contact tracing for flu, let alone common cold (which is caused by a bunch of different viruses)
Also, common cold mortality is extremely low.
Also, common cold mortality is extremely low.
Even if it's not the most important thing we could do, eliminating influenza and the common cold would be pretty fricking awesome.
To answer your Q, Op: yes. Provided Apple/Google doesn't remove the functionality in a later software update.