I have been involved in PSD diving since the mid 70's and methods have changed. We now bag them in the water with specially prepared body bags with mesh grommets installed to capture trace evidence. We did not use to do that.
Years ago I have seen victims brought up with lift bags and even worse, with grappling hooks.
One lift bag incident that was pretty infamous occurred when we were doing a recovery of a boating accident victim in fast running current in a tidal river. We had obtained a bunch of paratrooper inflatable life preservers. These inflated by a CO2 bottle and were worn on the parachute harness. They inflated in an hourglass shape and we found that by cutting them in half they made a great CO2 inflatable lift bag that had about 35 pounds of lift. They had handy brass grommets to attach cords to and stowed nicely in a pocket. When inflated they were a big, yellow and round. The victim was located but the current was making it difficult to swim with the victim due to the large amount of drag and they were being tumbled by the current. The divers tied the lift bag to the victims belt and hit the lanyard. Off the victim went to the surface, marked by the round, yellow lift bag.
What made it so bad was the crowd that had gathered to watch, and the diver had decorated his lift bag with a laundry marker and had drawn a big smiley face on both sides of the bag.