Maybe, maybe not. I'm not a fan of AI. I think the possibility of fraud is too high. One vial of goo looks an awful lot like any other vial of goo.
There's nothing but emotion to suggest he would have been a valuable producer, there's a lot of good stallions out there... At least their will be as long as AI and stored semen is not allowed. If a big supply of Secretariat had been stored, who'd ever choose anything else? Natural breeding does ensure a more diverse gene pool, and no one stallion can dominate the gene pool. Those are valuable reasons to leave it the way it is, though many breeds have given that up for sake of the economy of it.
OK, that's a good point. Although they could make an exception for one offspring (I guess once you allow one, you can't prove it isn't happening hundreds of times).