Some of this is getting silly.
Of course horses should be allowed out in public, especially on bridleways. Clue is in the name, really. And you cannot remove all risk from public spaces- unless you ban dogs, cyclists etc too. Or we could have bridleways which only horses can use, which seems a bit unfair and drastic to me.
Of course riders should make sure they are not over horsed and that their mount is reasonably well schooled. Horses should be reasonably expected to cope with everyday obstacles in most circumstances (although younger horses have to be trained somehow, but should be escorted by an older, safe horse in the first few instances).
That recognised, horses are not, and never can be trained to be, robots- even the most placid schoolmaster can be frightened by something. Some things are more likely to frighten a horse than others- common sense dictates that users of bridleways and other public spaces should take sensible precautions to avoid doing those likely to cause fright to horse when they see them. For example, don't let your dog approach a horse (inquisitively and with no aggression or otherwise) or put an umbrella up whilst a horse is passing. This is echoed in things like the highway codes instruction to pass horses wide and slow when in a car. It might be a pain, and horses on the roads should be fine with vehicles- but passing by them closely or at speed would be stupid and dangerous. So you don't do it. You don't do it to be nice to the rider or only if the rider is polite to you- drivers should do it because it is the safe and sensible thing to do and because presumably they don't want to be involved in an accident which may cause the injury or death of themselves, horse and/or rider or damage to their car!
By way of example, one dog I encountered recently bound out from the undergrowth almost under my horses feet. Think the dog was as surprised as my horse TBH, and I don't think it was attempting to attack us. Fortunately, the dog only got a glancing kick as my horse spun round and no serious injury to anyone, but it could have been so much worse for all involved. The owner had plenty of time to call the dog back as they saw us approaching- but as the dog was "fine with horses" they didn't think to recall it. I don't mind whether the dog is on a lead or not, as long as it is recalled and the owner in control. I don't think your dog approaching "inquisitively" but not to heel etc counts- dogs can be as unpredictable as horses and I don't know what your dogs intentions are or how my horse may perceive them approaching. For the sake of a few minutes inconvenience an accident could be avoided- so why not just do it?
I recall my dogs to heel and sometimes put them on leads when encountering horses. I also don't let them gambol up to strange walkers/children/cyclists etc- because I cannot be 100% sure what he will do or what those other people might do.
Of course riders should be polite when asking others to do something, but even if they are not surely the sensible thing to do when asked to recall your dog or whatever (even if asked brusquely or to do something which you don't see the point of) is to do it, as potentially causing/being involved in an avoidable accident is just does not worth it to make your point.