Do talent show programmes film lots of footage or is it set up?
I've shot live events for the past 5 years, and had a few friends get on (and actually pretty far in) both American Idol and The Voice. There is a series of auditions way before contestants ever get in front of a studio camera, let alone a TV audience. Most auditions are not filmed, unless it is recorded by the casting directors/producers purely for their reference, usually with a simple handycam.

Yes, some auditions are staged a certain way, but not usually scripted. The nervous shots and emotional cut-aways you mentioned are almost always staged, shot after the fact, and then edited in a way that lets the audience project meaning onto the shot. The exception of course being when producers know a big event is about to take place (e.g. put a cameraman in the green room right before the contestant goes on stage.)

At this point it becomes a matter of efficiently handling your media so producers can quickly call up a shot that, even if they haven't seen it yet, they know it exists. That's why digital intermediates on set are worth their weight in gold for "reality" TV shows.

If it's a live show, there are many cameramen all wearing earpieces, ready to take direction from producers on who/what to cover and when. Then you have a big team of people behind the scenes watching what every camera sees and punching up the appropriate shots to the audience. So in this sense only, yes they just shoot a bunch of footage. But a live streaming video is a bit different than recording to hard disc.