There are no "rules" that govern the Triple Crown. There are three races; each has different entry requirements. The Derby is limited to 20 entrants, and the ranking of those entrants comes in to play every year these days because owners want to be part of the Derby festivities. The Preakness is limited to 14, which occasionally comes into play--the Belmont probably has a limit, but it's the only mile and a half classic left in American dirt racing and I can't remember a time that entrants were squeezed out.
Each race has a different challenge. The Derby distance of a mile and a quarter is often a new test for these horses, and they are relatively young to face it--though it's true that many more TBs are born in January in the modern era, Man O'War's owner famously skipped the Derby because he felt it was too early to run a three-year-old a mile and a quarter. Especially with fields over 15 horses, a lot of horses who qualified for and ran in the Derby sit out the Preakness and Belmont after running poorly or running to exhaustion (often with poor trips or injuries). Two weeks later you come back with a shorter race, but usually some fresh foes (there's almost always a horse who loves Pimlico and a horse whose Derby plans were derailed due to injury or illness, at minimum), Then three weeks later, the Belmont, with a significant increase in distance (to a mile and a half), and not incidentally to a much larger track (there are some illuminating graphics in
this Bloodhorse piece). And the Belmont almost always has a few horses primed for that distance, who simply weren't rushed into the Triple Crown trail. Each of these races is a classic in its own right; winning all three is, and should be, a truly extraordinary achievement. It makes zero sense to suggest that horses ought to have to qualify for the whole series.
Joe Posnanski wrote a great piece on this.
If you've never watched
the full 1973 Belmont, you should. (And really you should go to the Paley Center in NY and watch the entire broadcast, including the part where the commentators question whether Secretariat can get the mile and a half.)
(Full disclosure: somewhere in a closet in my house is the poster I made of clippings of Seattle Slew.)