Remember under present theory on the evolution of the Horse, it evolved in NORTH AMERICA and than migrated to the Old World at the same time the Ancestors of today's Native Americans were migrating into the Americans.
With the introduction of the Ancestors of today's Native Americans into North America, several animals died out. The Camel, the Horse, the Mammoth, the Mastodon, the Giant Ground Sloth among others. There is a debate on what cause the disappearance of these creatures, the majority opinion being the Native Americans killed them off, with a minority opinion blaming the gradual change in the Climate. Given that Wild Horses have survived so long after being re-introduced during the 1700s I lean to the Native Americans killing them off but the change in the environment (which includes the introduction of Humans into North America Environment) was a factor.
But that brings me to the point I am making, the HORSE WAS NATIVE TILL 10,000 years ago. That is a blink in the eye of evolution. It evolved in an area like the Modern American South West. Many other Creatures in the same Eco-system evolved with the horse being in the Eco-system (Including almost all of the native game animals "threatened" by these wild horses). The horse is NOT a creature that the rest of the Eco-system had never had to address (like the introduction of the Chestnut Blight did to the Native American Chestnut Trees which has lead to the Chestnut tree very near extinction in the Forests of Eastern North America). The Horse is a Creature the Eco-system can handle.
Thus the Horse should NOT be viewed as an "Invasive Creature" but a restored one. I am for hunting them for there is no way you are going to see the wide spread introduction of Wolves into the same Eco-system. Thus some sort of predator has to be used and that is the Human Hunter, for it is only the Human Hunter who we can ever really control as to what creature he hunts and where. Wolves, Coyotes and Pumas will go after what is easiest for them and if that is someone's cattle such predators will do so. The Horse and its competitors among the wild creatures of its Eco-System evolved with Predators, predators that we can NOT re-introduce. Some thing has to do the job of Predator and that leaves the Human Hunter.
Thus while I can Not say the wild horse is an invasive species, I also see that the population of the wild horse must be kept in check and thus culls like this are often needed.
For some more on the Disappearance of the Mega-fauna:
http://www.bagheera.com/inthewild/ext_woollym.htm