There's a lot of generalization going on here. Like many other things, how clicks are handled is a case-by-case basis.
Sometimes only the conductor hears the click. Sometimes only the drummer. Sometimes conductor and rhythm section. Sometimes everybody hears it whether they want to or not, and sometimes the musicians have control over their individual mix (i.e. Avioms or Roland systems).
Sometimes the "click" is a clicking sounds, sometimes it's (like in the case of Radio City) the sound of somebody counting and occasionally subdividing beats.
The click track could be to line up pre-recorded vocals; pre-recorded drum beats; to prevent dancers or singers from complaining about shifting tempos (it never works); or because some technical element requires the same timing every show.
The click could be triggered by the conductor, drummer, percussionist, keyboard player, sound mixer, or even from the light board. It all depends on the specific need and the technology available at the time.
And to say that "most dance numbers have pre-recorded vocals" is disingenuous. Many do, most do not.