I have been told that wrap strap girths are designed for 4-wheel carriage shafts and race bikes on Standardbreds. Buggy shafts are not supposed to float, no weight on the horse. So the horse is harnessed with the wrap strap to keep the shafts quiet in the shaft loops while going along.
Thinking is similar on the race bikes, jog bike at the track. Weight is light, surface they go over is prepared, smooth. You do NOT want extraneous movement of shafts during times the horse is hitched. That movement could interfere with his speed, make you lose a race or cause a problem doing training sessions. So the racing folks also tighten the wrap strap around the shaft and snug it down to the strap buckle on the girth.
I am sure there is some cross-over into other breed fashions, ways of driving horses. The fine harness horses usually have wrap strap girths for less bulk, cleaner look. Not sure if the Thimbles that hold the shaft ends for stopping, are still fashionable at all. Again, vehicle is light, 2 or 4-wheeled, moving on prepared surfaces, allows less “free movement of shafts” during gait changes and halts, than floating shafts would. So even though horse may be carrying weight with a cart at times, shaft weight is not a big load or over hard going, rough surfaces or done for only short periods of time. Not usually going to sore him up. QH and Western breeds use this light kind of harness to their very light 2-wheel carts for breed show classes, have the wrap straps on their harnesses. Probably just copying other breeds in what they use for “driving” horses in the show ring. Some of the strapwork is so thin on big QHs it looks kind of odd, but IS THE FASHION they want to see.
Wrap straps are very common on “buggy” harness used by the Amish, sold by their harness makers.
I was also taught to bring the wrap strap up between horse and shaft, come over the shaft, down under the shaft loop, back up between horse and shaft again, and then over the shaft top down to buckle into the extra girth buckle for the wrap strap. Holds the shaft down and in place so the shaft loop is in the correct location. We never did the locking wrap on that strap, but there are bound to be numerous methods of fastening it. We do the locking wrap on the breeching holdbacks. If the footman loop is in the correct location on the shaft, I don’t see the girth wrap strap and the breeching holdback strap even getting near each other.
I will disagree that this wrap strap would ever be correct on a Gig. Someone may have used one to their Gig, but it was incorrect harness for the vehicle.
Gigs use a sliding backband on the correct, very wide saddle of a Gig harness. You might see a Gig hitched with a harness saddle using shaft loops (tug loops) that would allow some float. On the Gig harness saddle the more correct French Tugs will hold the gracefully curved, very hooked shafts in place. On correctly turned out Gigs, there would also be Tug Stops on the side/bottom of the shaft at the saddle area. The Tug stops will help prevent shafts sliding forward in shaft loops or French Tugs while stopping or slowing down, doing any downhill driving. No chance of vehicle hitting horse in the rump.