Whether the dog will complain depends on the injury, the dog, and the situation. It's the handler's responsibility to monitor the dog to prevent harm, if at all possible. Unfortunately, not all dog owners/handlers practice this.
If I'm hiking on rocky or icy terrain, I make frequent stops to check paws. This includes spreading toes to remove things that may have lodged between pads, and checking that nails have not worn down to expose the quick. I also bring, but have never needed, booties. If my dog's pads were not thick & tough enough to handle a rocky hike, then I would not bring them to a rocky area. A good ice crust or granular snow can sand their pads off, particularly around the edges.
I'm not surprised by Bobmak's experience on Whiteface. I've seen similar; and worse, a dog that collapsed from heat exhaustion and died soon afterwards. The dog was loyally following his ignorant, thoughtless owners at the cost of his life. Some people don't seem to even bother to look at their dogs. I've pointed out things like bleeding wounds, porcupine quills, bloody tracks, etc., to owners that somehow failed to notice these things. Never mind checking for dehydration, overheating, frostnip, and things that don't involve actual holes in the dog. I'll quit now before I go into too much of a rant.
