The "light cone argument" is equivalent to the light-speed limit, it's a geometric way of demonstrating it. It's not an assumption, you can prove it from tha axioms of special relativity --
Suppose you had a train moving faster than light, and had headlights on it, or a glowing guy sitting on top of it, or whatever. According to a stationary observer (in our frame), the train is moving faster than light, and the speed of light is fixed, so the train is always ahead of the light it emits. But according to an observer sitting on top of the train, the train is stationary, and light still moves at the speed of light, so the light is always in front of him and the train.
It's easy to see why this is a contradiction -- suppose there was some kind of high-tech wall at the end of the train's path. It's programmed to create some sort of catastrophe if it's hit by a train, but only works in complete darkness. According to the stationary observer, the train hits the wall first, before the light -- so nukes go off across the world, his family dies, and he ends up drunk, radioactive and homeless. But according to the guy on the train, the light hits the wall first, so it switches off, and nothing dangerous happens -- he looks around, waves at the stationary guy, who's completely sober.
You can't have contradictions like this in the real world, which is why the only conclusion that can be made is "the faster-than-light train doesn't exist".
(Technically, that's not the only conclusion -- we made the assumption in our analysis that time runs in the same direction for both observers. If you're willing to forgo causality, you can, in fact, have faster-than-light travel. That's why causality and locality are typically treated as synonyms in relativistic physics.)
Bonus -- why we like to talk about light cones: an example
"But isn't the order of things relative anyway due the relativity of simultaneity and all that? How does that not lead to a contradiction?"
It doesn't lead to a contradiction, because disagreements about simultaneity only apply to spatially separated events, which cannot possibly be causally linked -- as long as you don't have observers moving faster than light. The full proof is here: Light cones and causality, here's a quick snapshot:
This is another (but equivalent) proof, by the way, of "no observer's axes can cross the light cone", or "no observer can move faster than light".