Keep in mind that puncture vine, or Texas sandbur as we call it here, is an annual plant that thrives on disturbance. "Violent" efforts to control it merely encourage the germination of new plants.
The key to controlling it is to be persistent at removing plants before they have a chance to set seed, with the least amount of soil disturbance possible (a weed burner is a good idea if it can be done safely). Your driveway will be the biggest challenge because compacted bare soil is the ideal place for it. Seeds can remain viable in the soil for years so there's no short term solution but, if you are persistent at preventing the creation of new seeds, you'll eventually get rid of it.
One thing that won't work is pouring increasingly toxic chemicals (vinegar, kerosene, etc.) on the soil. If you ever visit the Trinity nuclear test site where they detonated the first A-bomb, you'll see puncture vine. If it can come back from that, it's not going to be bothered by vinegar or kerosene for long.
