As Chromes owner said, horses have to earn enough points to qualify for the Derby. Once you're in, that should be the field for the Triple Crown. No spoilers allowed.
Of course the playoff are a single tournament. The Crown is not, it’s 3 separate events chosen for their differences to test a horse’s complete ability.
Chrome’s owner is an idiot. The races have ALWAYS been open to horses that didn’t race in the previous events. Spoilers ENCOURAGED, it’s what makes the Crown a challenge. If it was the same field of horses in all 3 events there’d be Crown winners every couple of years.
Horses are different from humans. Horses don’t know there is anything coming up to work at, so they personally are not ready for races, the humans only try to make them so. And no one wants to hurt them, and cannot know really what is going on with them that might hurt them including “just not feeling perfect”. A human knows if he just cannot go on and can bow out, but he also knows if he wants to push himself to a set schedule that is required to get to regionals, nationals, etc. One just cannot do that with an animal.
Thus, there will never be requirements that horses go from one thing to the next. It is unfortunately one of the impediments to racing being popular these days.
The Triple Crown is a simple construct. Each race is its own entity and of its own importance. We artificially put importance on this “TC” being a string of events that just happened to become very important IN THEMSELVES. And that is why the TC exists at all - because people thought all of them were of high importance. It really did not exist until the ‘40s.