G&S – one teeny, tiny caveat. Racehorses are not asked to take any particular lead, and switch on their own as circumstances dictate. Direction is not part of anything to them, although they do know that if going the “wrong” way of the track, they will only be walking or jogging (at least, that is all they are supposed to do under normal circumstances). I have known jockeys who would canter a horse the wrong way prior to entering the gate, presumably to either loosen the horse or calm him/her down. However, they never cared which lead the horse took. They (the horses) will change during a race, sometimes more than once, but it is on their own, not because someone asked for it. It isn’t always comfortable for the rider when they do this.
Totally agree with everything else you say, but a lot of horses are taught with the use of the outside rein, and asking them to do it correctly can confuse them initially.
It is never the horse's fault