Translation:On air the investigator asked who can speak Korean.
No, the quotation remains in present tense. Think about it this way:
You) - "What is it?"
Me) - "Sorry, what did you ask?"
You) - "I asked what is it?"
The fact that you asked something happened in the past so that becomes past tense, but the thing that you're asking, i.e. the quotation, remains in the same tense it was originally quoted in.
You are right.
방송 simply means broadcast, emission
It was originally used as you said to mean a transmission of voices and images to the audience at large via radio & TV electromagnetic waves These being invisible to the eyes give the perception of "on-air".
방송에서 하다/방송하다 as used here really should just be interpreted as to make an announcement or to broadcast ...