I would add one case where spontaneous harmonizing is probably not a good idea.   That's when someone is introducing a NEW song or new tune.

Too often, I find when someone is presenting a new song that has familiar, common chords, when the group starts to sing along and starts adding harmonies from the get-go, nobody really learns the TUNE. So it ends up sounding like a dozen other songs that have similar chord patterns (e.g., Goodnight, Irene and Roll On Columbia would end up sounding identical).   
In that case, I think it helps if everyone tries to stick to the melody until people have got that down. Then they can add harmonies later.

But, as has been said, harmony (or melody transpositions) can be the only way you can sing a song that's really in a bad key for you.