People like them because they're horses, which are basically pets, and they might like that the horses might be like the ones eating nuts from your hand: http://hikearizona.com/photo.php?ZIP=513881
I remember hearing a girl or woman say she liked the tame burrows far more than the mean buffalo, back in 2000 in Wind Cave NP in South Dakota, because the burrows let her feed and pet them. Maybe that was Custer SP. Every horse takes away food for both livestock, but also the wildlife. Horses need better fodder, too. Cows, and the native grazers with multiple stomachs, do not.
The Feds don't want them because they are an exotic species with out predator control and probably without human management and if anything like the ones around Kayenta, they'll graze down everything and compete with the desirable stock, and also can end up rather sickly looking. They could harvest from time to time to keep the numbers the down, but I expect they want them gone. Like Salt Cedar, they're not without their pluses, but they just don't belong there.
Native North American horses went extinct, probably from humans, at the end of the Pleistocene with nearly all of the other mega fauna. The Sonora desert of today didn't evolve with the exotic domestic horse. Sure, there are tons of other things altered from 1492 in the west, and why pick this one? I don't know, but there is probably other information around if you wanted a greater expansion of their rational.