I've done this a number of times. The advantage is speed, especially with a loaded touring bike or a bolt on IGH wheel.

I used to lead bicycle tours and it was SOP to do this using a band-aid if, for example, there was a train to catch. I have my fastest rider switch places with the person who had the flat (within limits of bike fit), send the group ahead, fix the flat and the two of us would race back up. Being able to do things quickly was often the difference between catching a train or ferry, or waiting hours for the next one.

BTW- you can't use band-aids as emergency patches any more. The strips are micro-perforated and won't hold air.