Correct on both accounts.

In the studio you want to capture just the voice so you isolate the singer from the rest of the band, whether live or prerecorded.

Hearing the band while on stage is notoriously problematic. You either just hear the instruments/amplifiers, in which case you only really hear the closest ones to you, or you have stage monitors ("wedges") that add to the already high volume, or you have the sound engineer feed your preferred mix into your in-ear monitors where you can have reasonable volume and prevent damaging your hearing.