I'm not parsing this. Are the pointy pieces of wire every six inches or so not barbs?

A barb typically extends off a "pointy piece" to prevent it from being removed easily without further injury. For example, some fishhooks and arrowheads. The fences aren't designed to poke and then stick in the flesh.

Obviously you wouldn't want that for livestock, but this razor wire fits your description:

https://shop10303.ldiiaceh.org/category?name=cbt%2060%20razo...