You have it backwards. Back in the 80's and 90's when I lived in Santa Fe, you could hike up in area of the ski basin and never see a coyote. The parking lot to the several trails was full and people (and dogs) were frequent on the trails for the first couple of miles and less so as you got further in to the areas like Puerto Nambe, Santa Fe Baldy and Lake Katherine. Great for day hikes and weekend backpacking. Coyotes have invaded the area and are now as much of the varmints as the rodents you mention. They now threaten the fawns and and maybe the young of the bighorn sheep on the east side of the mountains. Dogs and (maybe small humans) are just as much a prey animal to them as other small or young mammals.
Up until the 80s many people trapped and shot varmints all over NM; coyotes, fox, bobcat- furs were worth some money and regulations were few. Fur prices plunged, regulations grew and nearly everyone quit trapping and varmint shooting it just wasn’t politically correct- anti fur people caused money to dry up and too many laws to break. People were keeping the numbers down but other people changed that. Now we have large groups of people that don’t want the “cute” varmints killed, instead they want to feed them- yes we have idiots around here that put out dog food for coyotes, skunks. I bet idiots around Santa Fe are doing it too and there are more idiots there. Population of varmints has exploded and people are moving more and more into their habitat.