No, not all vets are good at spotting horses who are sore on all four feet. I have direct experience observing that. Unless a horse hits some rock or big gravel in the trot out, it can be hard to spot, not all do that "walking on eggshells" thing. I remember a couple at one of my rides who barely passed the final vet, most of the ride was sandy and/or decomposed granite, few rocks. They were riding barefoot, and proud of themselves for completing the LD barefoot, they couldn't "see" how sore their horses were. They've learned a ton since then and are now somewhat authorities on the barefoot plus hoof protection thing, but that day their horses definitely paid a price.
And poor saddle fit often does not show up symptomatically until way after the ride. Poor saddle fit can result in lameness pulls from the horse straining something else while trying to protect his back.
I think having a discretionary "hoof protection required" rule is a needful tool for a ride manager. It doesn't say shoes vs pads. I know of no rides that say padding is required, but many recommend it.
I know they have a right to make a rule about bare hooves, but I don’t understand the need for it any more than the need to require padding. Does any ride “require” padding? Some rides are particulary rocky and there are always horses pulled for lamness due to the rocky terrain, but the pads are recommended, sometimes strongly, but not required. I suspect it’s a desire to protect the horse from discomfort, but some bits are pretty darn uncomfortable, saddles cause pain, etc., and the horses are just pulled if they are determined to be suffering in any way. If the barefoot horse is ouchie, it’ll get pulled, right? Then the owner would know they need hoof protection, just like the owner using a poor fitting saddle will know it’s time to get a different saddle if their horse is pulled due to a saddle sore or sore back. It puts the responsibility onto the owner where it should be, IMO.