Really, a lot of people don' know. To most people, horses are expensive, beautiful, slightly mysterious things that we go past on walks.
There's a lot of 'folklore' about feeding horses apples, carrots, polos, lumps of sugar etc., so people assume it's OK. I'm not saying that it is OK, but people wouldn't think differently about patting or feeding your horse as compared to a friendly dog.
I think DO NOT FEED THE HORSES is fine, but it's a bit shouty, and doesn't really help people to understand why. Most people DON'T know about equine digestive systems or laminitis, they just see the horse there in the field that they pass all of the time. I wonder even now - is it OK for me to give a horse a handful of long grass (not cut grass) that's growing on the footpath side of the fence? That's what I have done in the past. I don't know whether that's OK or not.
You shouldn't have to, but if you keep your horse on a popular route, why not make a sign that say, 'Please don't feed our horses because... they have sensitive stomachs and can't be sick... we have to monitor exactly what they have... they can get special horsey illnesses that make them lame and could end up with them having to be put to sleep..." Maybe also let people know whether or not they can pat the horse.
Please don't assume that anything at all to do with horses is 'just common sense', it isn't. Most people are well-meaning I'm sure, and at this time, yes - probably looking for something to do. Why not tap into their fascination and help them to learn more about horses in the process?