On the one hand, I sympathise, OP - I've known some very... worried... riders who didn't really seem to have much control, or confidence in their ability to handle normal situations. Being asked sharply to lead my dog (who is "well horsed" and very obedient - and would've been called back at the sight of the horse, and held by the collar if they were anywhere near us) would irritate me too.
People can be rude when they're nervous, and I sympathise with that too - but it doesn't really make it ok. And a (controlled) dog is not an unexpected thing to meet out hacking - it does make me wonder what they would do if they met some of the things I've met out and about. I had a deer jump into my horse's face once - that was a special moment - and have met big groups of classic tractors on a track no wider than they were on a very young, inexperienced horse. We've also met herds of alpaca loose (escaped!), landing hot air balloons and various emergency vehicles with sirens going. If you can't cope with a dog walker, you probably do have bigger issues being out and about.
That said, one of my horses is aggressive towards dogs and may attack one which that came under his feet. He would if he were loose (he has in the field) - under saddle he has thus far always retained his manners, but it's not a chance I would take if I met a dog walker who did not appear to be in control of their hound. The other one is very good with dogs, but would be a silly bugger if one lurked in the undergrowth and he could see some hidden predator in there. I'm not usually stroppy with people I meet who alarm him (I know him inside out and can deal with his antics - so I'm usually laughing and apologising for his behaviour!) but it's always possible you'd catch me at a bad moment.
I would also like to point out that the horse isn't the only dangerous animal in that situation. If the dog "nips" my horse's legs, there is a very real risk that he could pierce a tendon or even a joint capsule, which could potentially be a career ending injury - even through their boots. When you're riding an animal worth thousands (even tens of thousands) of pounds, who you've spent the last 5 years training to a fairly high level through blood, sweat and tears, you can get a bit funny about strange dogs around their feet!