Perhaps I’m wrong here, but when it comes to vocal teaching I usually see the vocal coaches using a piano; not all the time, but when ever an instrument is in use it’s always a piano from what I see.
Probably because piano could be seen as a "universal" instrument to compare others to.Also, it can be seen as the "cleanest" instrument. It has no special timbre/tone, and it is a plain, simple view of all the notes in the western musical scale.
Basically, it’s all FLAT OUT right there for you, not in a more complex form such as guitar,trumbet, etc.
No air is needed, and you can reference easier, as it’s always been the traditional instrument, of sorts, as mentioned before.
These are just things I came up with now, they seem pretty logical though, yes?