Back here in California, where so many H/J teachers came from the strong western traditions (Jimmy Williams being the most prominent), having your students run barrels bareback in that time period doesn't seem out of place at all. Very few lunge lines out here at that time, either. Nope, you just took away the saddles (and for the advanced, the bridles), closed the gate to the arena, and let everyone work it out for themselves.
Huumph! LOL I did clinics with Hugh Wiley and a once-in-a-lifetime lesson with JW waaaay back then, and Wiley lunged me over fences w/o stirrups or reins or ran me through a Hitchcock pen (depended upon what facility the clinic was at), and Williams made me drop stirrups. Neither suggested barrel racing to improve my seat. LOL
Most trainers that I saw in southern California didn't lunge - you set up a gymnastic and the rider aimed and dropped the reins. A lot of places had jumping chutes.
Oh yeah, we did aim, kick and drop reins. ROFLOL Had to have a honest horse, though, when you don't have a Hitchcock pen. We also did drop reins, remove jacket/sweater while riding down the line of jumps.
My husband and I watched it last night, and it really was entertaining. I thought it was really quite well done. My husband started out poo-pooing it and groaning when the father decided to get into horses. Actually he was shouting "No! Don't do it!" but he got into it in the scene when the father rides the horse bareback. By the show jumping at the end he was hooked, and quite impressed with jumping.
He even asked if I had ever ridden over jumps and started studying what the jumps were made out of. I think I might get some ground poles and caveletti out of this yet
Marriage: an on going experiment to prove there are at least two ways do do everything.