You could use a synonym like Luftstreitkräfte (or Luftstreitkraft, German Wikipedia seems to use both singular and plural and it's clear to me which one is correct) like the other comment suggests. To me, it just seems weird to still use the same terms as the Nazis, but it makes sense considering the whole ”no denazification” thing, I guess.

there shouldn't be a US army

I agree.

You just mean the word army, right?