That’s pretty orthogonal to whether or not the political leader of the US should publicly say that looters should be shot.
...neither of which is an outcome in this situation, and looting is not a credible treat to people's lives.
No it isn't. Legs are hard to hit compared to center-mass, and the only reason you should be shooting at somebody is if you need to because they are an imminent threat; therefore you should be aiming for a part of the body that you have a higher probability of hitting and that, having been hit, has a higher probability of effectively stopping the threat.
First, are there circumstances in which a citizen could apply lethal force to protect life and limb? Obviously yes; shooting someone trying to light your house on fire is certainly something that is both plausible and plausibly legal.
Second, is it appropriate for the political leader of a nation to imply that people committing a crime will be shot for it on the street? Not "we will deploy the national guard to provide safety and security to the citizens" but "looters will be shot". I think the answer here is obviously no, that is not appropriate.
I will also add that this is also a case of tensions boiling over. While that doesn’t justify the arson, meaningful reform to defuse long standing tensions would be a wise move.
You’re right, everyone’s on edge, which has people acting funny.