Do animal trails use switchbacks? = No
How do you prove a negative? You can't, but consider the alternate question; "Why would an animal trail use a switch back?"
Goat climbing cliffs maybe, but that is not so much of a trail as a search for good ledges
All other trails are made by herbivores traveling between locations. These trails change with washouts, and fallen trees. There is no good reason for an herbivore to make a switch back. A long gentle climb around the side of a mountain or hill, has more advantages in energy savings, more options for browsing, and less worries about predators overhead.
Carnivores don't make trails, they follow herbivores, or hangout along the trail.
Omnivores (bears, pigs) Much like herbivores, don't have a good reason to make a switch back either.
Trail builders have four foot drive, and are never hauling heavy loads. They have no concerns for trail maintenance, or proactive erosion control. Much like people on a trail, they will tend to Short-cut any bend.
Navigating Switchbacks to Prevent Trail Damage
A switchback is a trail up a steep hill or mountain that is like a zig-zag pattern instead of a straight trail. The zig-zag pattern protects the hill and the trail from excessive erosion. Trails that go straight up and down steep hills don't stay nice trails for long. Erosion turns those trails into gullies because water moves faster down steep straight-aways and it hollows out the trail and washes all the soil and vegetation down hill.
Switchbacks help prevent erosion because they help keep the trail at a consistent gradient. Trail planners build other erosion control devices like rolling grade dips into switchbacks also. Switchbacks are also good because they make it easier to climb steep hills.
"Short-cutting" a switchback is bad trail etiquette because it kills vegetation and loosens soil creating a new trail straight up and down the hill, which will in time get large and hollowed out from erosion. Do your part to keep our trails beautiful and control erosion. Walk softly, Leave No Trace, and stay on the designated trail!!
In summery switch-backs are tools for long term maintenance of a dedicated and maintained route between two specific points. Trail making animals don't have any motivation to create them, and every motivation to exclude them.