I wouldn't take my dogs out with me on a bridleway because although they are well trained and have good recall I cannot be 100% sure they will not do something unexpected to another horse/dog/cyclist/runner/walker etc. I know some people do, but I imagine they are not the people who would be asking other dog owner's to put their dogs on leads.
I happily let my dogs loose when I'm riding on my own land though, in the knowledge that I they might do something daft. However, as it is my land, my horse and my dogs I would only have myself to blame (and only me or my animals would be affected). Likewise, I don't mind if I come across a dog not on a lead as long as it's owner has it to heel/clearly under control. The op said she would have to "dive into the nettles" to retrieve her dog, which does suggest the dogs are not exactly under control and she is relying on the fact that the dogs have not approached/barked at/attacked a horse before. Which is, in my view, irresponsible.
I agree that riders have to take some responsibility too- they should ensure that they are capable of riding the horse they are on and that the horse is reasonably well-schooled. However, no horse can ever be expected to never spook/kick out etc if frightened- even the most docile, well schooled and sensible horse can act this way if frightened. Young horses need experience to become accustomed to dogs, they don't just come "ready made" and they are not robots. If asked in these circumstances, I really would expect an owner to either get their dog to heel or sit, or put them on a lead- for everyone's safety.
To those suggesting dismounting- that would, in most instances, be far more dangerous as you have less control on the ground than riding in most circumstances. I would not advocate getting off every time you see a dog walker, it just makes the horse believe there is something to fear.
I find it astounding that some dog owners would risk injury to their dog, the horse and rider and potentially themselves for the sake of a few minutes of their time or to make a point.