2y old foundation
 

Raced as a 2-year-old
This is the granddaddy of all qualifiers. At least one race at age 2 is essential if a horse is to win the Derby. Only Apollo in 1882 won the Derby without racing as a 2-year-old; only one Derby winner (Fusaichi Pegasus in 2000) since 1967 raced only one time.

Why do trainers keep trying to beat this? And, why do betters keep betting on them?

Just like must have a race within 4 weeks of the Derby or no RAN in the Dam Sire line, until Barbaro...

Most don't try to win the Derby with no 2yo races, that's most of the reason why...

It will happen, takes the right year and horse, but it will happen.

so seldom seen, since horses so seldom skip their two year old season.

I believe that only about 30% of any given crop runs at 2. Racing at 2 seems so ubiquitous but in fact its a relatively small bunch running. That said, it seems like winning the Derby is all but dependant on some experience at 2.