OP, just a thought, could there be a stray online review out there describing your business as 'dog friendly'?
We love taking our dog places (she loves it too, and so do most members of the public going on how many people want to pet her) and we usually check first if she can go in, but we did get caught out a few times in the early days because we'd assumed 'dog friendly' meant inside and it didn't. Still wouldn't have 'stormed off' though.
We'd never leave her unattended outside anywhere, she'd be a prime target for a dog-napping. We could leave her at home, and do if we're going somewhere she can't but we would we unless we had to? In a straight choice between lunch somewhere she can go and lunch somewhere she can't we'd go to the former every time.
As for someone who said you wouldn't take your cat, no I wouldn't, he hates being in the car, and I've totally failed to train him to lie down at my feet and go to sleep while I eat my lunch, he also gets nothing from meeting new people or animals, is upset rather than excited by new places, and given a choice would much rather stay home. I also wouldn't let my dog roam the streets unattended, or expect a snake to live in a duck pond, but there you go.
I think it's up to individual businesses to decide really, and customers can go elsewhere if they don't like it. People are BU for kicking off about it (and for not reading the sign), rather than going quietly elsewhere, but OP is BU for expecting that members of the public are anything other than arseholes.