The laws are more nuanced than you suggest. Here is a government website [1] that mentions dogs and says “Dogs must be under control at all times. Dogs can harass, stress, injure or kill wildlife; annoy fellow hikers and introduce disease. Some wilderness areas require dogs be leashed at all times.”

I have recently been in Mount Baker’s wilderness where dogs off leash away from trailheads are fine and also in Mount Shasta wilderness where pet dogs are not allowed even on leash. There is latitude for local tweaks to the rules in specific wilderness areas across the US. The US also has vast swaths of BLM and other lands where all kinds of fun recreation (hunting, dogs off leash, OHV’s, and all kinds of other things are allowed).

[1] https://www.fs.usda.gov/detail/psicc/specialplaces/?cid=stel...