To the OP: no, you're not being a narrow-minded jerk. Just preaching to the choir, perhaps? ;-) We dog owners hear these complaints a LOT, and it makes us angry because we realize enough violators could end our ability to bring our dogs camping.
My only suggestion: instead of brandishing a hatchet, which could get you into trouble - better you would keep some bear spray handy; or even an "ultrasonic repellent device" ($25 from Amazon) that makes an annoying noise that only a dog can hear and would chase away most pets (can also work on the neighbor's barking dog, if you feel like getting a bit of revenge)
The sad truth is that there are a LOT of people who feel their dog(s) have a right to run freely, whether in a campground, or a beach, or a park, etc. I have encountered those in the past: my former Cattle Dog was terrifying-looking enough to convince owners to call their loose dogs back after I explained that he would consider a loose dog in our "space" to be a dire threat. But now I have foofy-looking dogs; so I carry a small canister of Mace. Just in case....