I agree with the rule and Shadow Leader has summed it up perfectly. Jockeys are no more aware of any internal damage to a horse, whether it's sustained a hairline fracture which will open up during further riding (but which will show up the next day with a little heat and swelling in the affected leg), or even whether they themselves have sustained an internal or head injury. Riders can clamber back on feeling a bit sore, and, not being either vets or doctors, not realise that they're on the way to a serious internal rupture, etc.
Never mind KAUTO STAR - Ruby's climbed back on lesser mortals for a lift back to the stables, rather than walk the horse back. I mention NO NEED FOR ALARM, a huge mare of Paul Nicholls, who fell with Ruby twice at Plumpton. On one occasion, her lass retrieved her from me, holding her after she'd crashed through the rails at the Railway Entrance, no doubt spying the big gates as the way home. On the other, Ruby remounted her and cantered her back. She was lame next day. Was she going lame at the time, did cantering back make her condition worse? Who knows - but certainly Ruby didn't.
Vets and doctors are the only people to determine whether a horse or rider is fit for further work. Riders, of varying abilities and degrees of horse knowledge, aren't.
Horses who decant their riders as they're on the way to post or at the stalls aren't affected by the remount rule. It's horses who crash during actual racing (i.e. at racing pace).
There'll be mutterings into their beards by reactionaries who'll cite Sir Rupert Raddlestone-Popinjay riding his chaser to glory after three falls and remounts in a chase in 1923, I'm sure, but these are modern times where, one hopes, modern thinking is gradually inching its way into racing's psyche. All of racing - not just the more innovative Flat, but grizzled old jumps, too.