Regarding clamping the rod, using rubber, wood, leather, or something that would deform enough to fill the threads so a clamp or vice would work have usually worked for me. Once, when it did not, I actually split a nut of the same size/pitch with a bandsaw and was able to clamp it in the immediate vicinity of some thread damage on threaded rod and it worked to hold the rod sufficiently so it did not spin. I also used that method to chase the threads sufficient to clean them up to get what ever was below the damaged portion off.
However, I've found that with some cheap chinese threaded rod-using tools, the weak link was the threaded rod itself. I've actually upgraded a couple pullers that I use frequently with some domestically sourced threaded rod. Sadly, they're usually metric threaded so a little more challenging, but the forged or cast components did not seem to be the issue.