As has been said, yabvu. Horses, just like dogs are unpredictable and any situation can turn very quickly. My horse is very good with dogs and regularly has a doberman in his stable with him without batting an eyelid. However, if I am out riding, I will always ask someone to put their dog on a lead, as that is the safest for both parties. I don't know your dog, you don't know my horse. My horse may be feeling a little twitchy and spooky on that day, and anything could cause him to jump, and I would hate for you, myself, your dog or my horse, to come to any harm because you feel your dog is completely trustworthy and will not listen to me, as a rider, asking you to simply secure your dog whilst I get out of your way.
Being offended by the rider not saying good morning to you is silly, as the rider may have been concentrating on getting the horse past the situation. Or she may have been having a bad day. Don't judge all riders on one experience of someone asking you to put your dog on the lead and not saying good morning.
Also, if I got off my horse to lead him past an issue, the likelihood is if he were to spook then I wouldn't be able to keep hold of him. They are big, strong animals who have no issue in using their strength against you when it comes to self preservation. I have much more control on board than I do on the floor when it comes to a scary situation.
99% of horse riders only want to get out of the situation alive and safe, with as few problems as possible. Being stroppy about having to put your dog on the lead for a minute only makes it more difficult for everyone. We are also, generally, very pleasant people, and I for one always make sure to thank cyclists and dogwalkers who stop to let me pass, and always say hello to people I meet when out and about. However, sometimes this isn't feasible and getting upset about it will change nothing.
This is Coco's sister by the way 