Here's a fact I did not know in relation to Barbaro not being used for artificial insemination: It's against the rules.

The Jockey Club has never allowed artificial insemination, or AI. Vials of frozen sperm are easier to transport and dilute and can impregnate more mares than live cover [i.e., intercourse --kb], so AI could produce a glut of thoroughbreds born from popular studs and mares. Some breeders believe this could result in the overproliferation of offspring from particularly desirable studs, and limit genetic diversity. If too much of a thoroughbred's sperm were available, it would be less rare, and perhaps less valuable. (It was estimated that Barbaro, who was 4 years old when he died, could have commanded $1 million a year had he recovered enough to mate. His virility was so valuable that his owners had it insured.)

The article has more facts about horsebreeding than most of us want to know. But it's pretty clear that this is an attempt to control supply and drive up prices for the most desirable studs.