Burning doesn’t really work, but it is quite satisfying!
Perhaps you can spray enough, but I haven’t gone that route because i don’t want to expose the animals or kill the other vegetation. So I’m no help there.
What does work, is going out with a hand trowel and digging up the whole plant with the tap root BEFORE they flower and definitely before they go to seed. You’ll need some serious gloves because even before they flower odds are good that when you grab the base of the plant there will be at least one intact goathead sitting right there to stab you. It’s so much work, but you’ll get so fit with all of that squatting and digging and pulling!
Flip flops are also amazing at picking up the goathead seeds. There has to be some way to build that concept out into a goathead collecting roller. You should develop one. I’ll buy it. It wouldn’t even have to be that good. Just helping pick up the really sharp ones would be amazing.