In the picture the lack of any horse equiptment on the horse, such as a halter and lead strap, link strap, breast collar, surcingle, valise, saddle bags or crupper and the rider not having any spurs has nothing to do with, is this or is this not Gen. A.P. Hill. A note here on Hill's presentation spurs, the ladies of Columbia, S.C. had silver spoon and forks melted down and the company of Kraft, Goldschmidt & Kraft cast a fancey pair of eagle head spurs inscribed to Lt. Gen. A.P. Hill. I doubt he used these on a daliy basis. He probably had a common pair of spurs for when he was in the field.
As for the snaffle bit they were used by everyone from a private to a general and more likely used by higher ranking officers since they owned or knew their mount very well. A soldier used no more of a harsh bit than necessary to control his mount. Yes some horses it may well have taken a military high port #4 bit, but on the other hand some horses may have been easily handled with a hackamore ( bridle with no bit). The common snaffle bit would fall in the mid range of comfort to the horses mouth, but it all depends on the horse and the fit.
I worked for over twenty years as a wrangler, an occasional stuntman and historical consultant in movies that used animals, mainly civil war and westerns, and when we were off the set working our horses, mules or oxen, we kept it simple by not wearing or carrying all the extra equiptment on the animals unless we were training them to a new piece of equitment. Like the man in the picture he may be a stripped down version of what he looked like in the war. Too bad he doesn't have the three stars on the lapel of his coat like Gen. Hill was known to wear. I would like to see a mounted profile picture of Gen. Hill.
And oh yes, you only tie horses by the reins in old western movies or not at all, so the wrangler can catch it when it runs off.