Whoever said that Johnny Cash was a great singer. He was popular, he was talented, pleasing sound, but not really a technically great singer.
If you don't see your fave singer breathing before each phrase, trust me it's because they learned how to hide doing it.
Back to the subject... your instructor is trying to teach how to sing properly with technique. I think there's something in singing technique that has you fairly undetectedly inhaling through nostrils before each phrase, as opposed to gulping or gasping audibly.