The default is that what's said is protected.
The question should be "why is this speech not protected?" Does it incite violence if no violence occurred? Was it some kind of targeted threat or just impotent rage?
This very obviously doesn't apply in this case.
1A protects you from laws or actions that prohibit your speech, they don't protect you from people asking you not to engage in that speech.
As I understand, other SCOTUS rulings on chilling effects all pertain to political speech that were otherwise clearly and unquestionably legal.
I think we’d need to have more information about the process and procedure that took place here to determine whether this letter was sent in bad faith or not.