A.I. is possible with horses & in fact is pretty much the exclusive method used with standardbreds. It can be used with thoroughbreds as well, but under Jockey Club Rules the foal cannot be registered & therefore could not race. Proponents of A.I. usually make arguments along the lines that it is safer & more "advanced". Those opposed often say that the natural way is the best way to introduce new breeding strains, thereby avoiding what used to be a relatively common situation in harness racing, namely all or most of the runners have the same sire. Conspiracy types say that a few prominant & wealthy families keep control of the best bloodlines be limiting the no. of times a stallion can be bred. In the movie you mention, my recollection is that Rourke's sidekick secretly stole some of Secretariat's sperm so that a mdn which looked like he had avg breeding was actually sired by Big Red, which didn't make a lot of sense at the time, as Big Red's problems in the breeding shed were actually what led to the so-called "fertility rollover" provision in stallion syndications.