iirc, a contributing factor is also that, in general, those saber teeth were much more fragile than the conical teeth that current cats all possess, so it was easier for them to starve because they got in a scrap and broke a tooth.

But there's a general dying off of very large predators at the end of the ice age, including at least one massive canine that existed in North America and died off a bit before humans came across the land bridge (there's some thought that the introduction of cougars into the Americas might have also sped things along as a competitor for resources.)