It is not just Fauci, Birx, etc, all other countries' government, regardless of governing systems, are looking at this number to make/change policy.
Policy can change R0, R0 in turn can make gov change policy.
It looks to me (and many other people) the lockdown policy in New York failed to lower R0 significantly enough. So NYC actually would be safe relatively say if the lockdown had been lifted. But, for the rural small towns and villages, the lockdown indeed could have lowered R0 and the equilibrium therefore could have been reached under the current lockdown policy. But, if they loosen lockdown policy, R0 will go up again and it will spread again. This is the fundamental flaw of the lockdown policy because it cannot be loosened, otherwise it will spread again from an imported seed. This is the epidemiological reason why I don't like this lockdown in first place. It makes America a hotel California, you can check in but you can't get out. Of course economic and social reason is another one I don't like lockdown.
The bottom line is Fauci, Birx, etc, are stuck in this situation. They have to monitor and calculate data very carefully to make policy change. Say, after you loosen some policies and R0 in your area go up back to 0.8 from 0.6, policy makers may think this rising is ok because R0 is still under 1. If it goes up back to 0.9, they will become nervous because R0 is close to 1 and they may get ready to tighten up the policy again. If R0 goes back to 1.1, meaning the virus start spread again, then they will definitely have to issue lockdown again. I don't like this. Again, I like Sweden's approach, once for all.
The other two numbers considered by your brother are also policy policy dependent, death rate (severity) ad case number. Although more and more evidences show the death rate tend to be lower than 1%, I understand Fauci, Birx, etc can't really make policy change based on these evidences. As for case number, I think I and you agree with your brother's rational.
Link: https://en.wikipedia.org/wiki/Basic_reproduction_number