During our camphosting over the past few years, I have seen picnic tables chewed on, small trees downed, holes in excess of a foot dug in the ground, and lots of other damage.
I have personally filled in these holes, changed out seat boards on the picnic table, and removed damaged trees, so yes, a dog can do quite a bit of damage at a site, if not properly supervised. Sometimes, even if supervised.
I don't think a $25 deposit is too much to ask. That way, if the dog does damage to the site, the responsible person pays it and not everyone that uses the park afterwards.