Why doesn't the media ask the candidates running for President whom they would recommend as members of their administration?

It’s way too early for the candidates to commit to these decisions. First, it’s too early to know who will be able for Cabinet positions. We wouldn’t want a candidate to name someone of lesser quality because a better staffer isn’t yet available. It would also be embarrassing for a candidate to name someone now, who then has to announce they won’t be taking the position because their personal circumstances have changed.

On a more practical and unpleasant level, there are still a lot of deals that have to be cut, and the Washington elite don’t want to commit to a position in a candidate’s Cabinet, only to have that candidate not get the nomination, and the rival candidate has filled all of their spots. The candidates also don’t want to commit a spot away now when they might be able to cut a better deal later, say, you throw your support behind me in this state’s primary and you’ll be Secretary of Energy.

And the candidates want the spotlight on themselves. Any announcement of other figures would diffuse the spotlight. Simple as that.

The media may not feel this is relevant at this time, for reasons stated above. Also it is often assumed that High level people involved in the campaign and people who endorse candidates may help in shaping or become part of the administration.