Other answers seem to address the use of ground anchors, but I think you're asking specifically about having a ground anchor with the belay device attached to it, and not attaching the belayer.
I've worked at gyms with this system, and the main argument is that the belayer doesn't need a harness. It's common in gyms that draw most of their revenue from children's events that the chaperones/parents won't climb at all. If you can get away without renting them harnesses, it saves time and reduces the amount of rental gear you need to dedicate to the group. This setup also allows the gym to have the belay devices (usually Grigris) semi-permanently attached to the anchor with a maillion so that inexperienced belayers can't set up the device improperly.
Many counter-arguments can be made, but the primary issue that arises is that the system breeds complacency. Since the belayer isn't directly attached to the system, they can be less engaged, and stop paying attention (or wander off..)