There's no strength in the term "Airman" as uniting everyone in the USAF - Because (unfortunately) we have a rank that is the same as what we are.

Airman Basic = E-1
Airman = E-2
Airman First Class = E-3
Senior Airman = E-4

There's no equivalent in the other services, such as "Sailor Basic," "Soldier First Class" or "Senior Marine."

Based on my experience - It's more common in the USAF to describe personnel as Officers, NCOs and Airmen than it is to describe everyone as Airmen.

Then again, not a lot of attention has been given to this subject because the USAF had lost nearly all focus on military tradition.