Time doesn't slow down when you approach the speed of light

Correct, but only from your perspective. To other people you've slowed down, but from where you're sitting (or careening through the cosmos at the universal speed limit) everything happens just as fast as it normally does.

the theory we're using to describe much of the universe is based on a bad premise, that the speed of light is constant.

Quasi-correct. "The speed of light" as we think of it in physics is actually the speed of information, which dictates how quickly changes can propagate outwards (or put another way: how quickly you can know about something happening elsewhere). We refer to it as the speed of light because photons move at that speed in a vacuum due to having no mass and thus moving at the fastest possible speed, but things like gravitational waves also propagate at that same speed and have nothing to do with EM radiation. However, the speed of information doesn't change; it's a hard natural law with no known exceptions.

Physics in general is cheating for this thread though, because the answer to what makes stuff happen as we understand it is a giant metaphorical mass of "I 'unno." The Standard Model, relativity, quantum mechanics, string theory, etc all have giant gaping holes in them that other models can often fill, but cannot be properly combined in any way that we've tried so far. They're still correct enough to base your entire life around without any worries, but there's always that last 0.01% that amounts to the margins of old maps reading "Here There Be Dragons".