OP I find some (not all) horse riders to be quite rude and arrogant. If I couldn't control my dogs I wouldn't take them out. Think the sam should apply to horses.
You haven't described horses being out of control though, have you? In fact, you have been very vague. All we know is that riders repeatedly ask you to put your dogs on the lead when passing you. And other dog walkers also sometimes ask you to put your dogs on the lead. Although in both cases, you're not sure why.
I think it sounds quite likely that you're a bit unaware of whats going on around you, a bit slow to react, and I doubt that horse riders and other dog walkers ask you repeatedly to put your dogs on the lead for no reason. They see an ineffectual dog owner, make a quick calculation of the risk (which you seem unable to do) and ask you to put your dog on the lead. I'm sure both groups pass loads of people with loose dogs and its doubtful they ask all of them to put their dogs on the lead, so applying logic to it, it seems that you are the common denominator.
I find some (not all) horse riders to be quite rude and arrogant.
Because they ask you to put your dogs on the lead when they pass you. Perfectly reasonable thing to do.
I've been bitten by a dog which was loose when its owner was walking it. If it had bitten a small child or a horse's leg, it could have caused terrible injuries. Just stop being so selfish, and do what people are asking you in a public space. Drop the attitude about people being arrogant and rude when they are being perfectly reasonable.