As a blind-leading-the-blind layman answer, if you shove a car forwards it will move slowly (ignore air/tyre resistance and such).
The same strength shove hardly moves an artic truck, it wobbles a Ford F-150, it pushes a sedan, it rolls a small sports car, without air resistance and axel friction, it’s only the mass which resists your shoving and determines the speed. and you can imagine waving a magic wand over the car to lower its mass then the same shove pushes it faster and faster. At the mass of a ball bearing the car pings away. Like a tennis racket swing can’t move a car but can move a tennis ball at 100mph.
Until there is no mass left, at that point any shove and it skitters away enormously fast. As mass drops the shove pushes it faster. As mass goes to zero, it goes “infinitely” fast.
There isn’t any mass left to remove, so the only way you can speed it up from here is to push harder instead... but now as soon as you touch it at all, at any pressure, it skitters away and you aren’t touching it anymore so you can’t push - damnit. The first inkling whisker of a skin cell touching the car and it’s off, the time it takes to “wind up” your shove power is too slow, the massless car is 300,000m away in a thousandth of a second. You can’t keep your hand on it, let alone go faster than it to provide more pushing force!
In fact the only way you can move your hand fast enough to keep up with it, so you can keep touching it long enough to give it more push, is if your hand also has no mass. And now your massless hand can’t be pushed by your real muscles for the same reason, once stuff has no mass you can’t get a grip on it. So pushing harder is out.
Maybe you can push it faster in a non-mechanical way like electromagnetically? Ok, you send some magnetic waves after it. Photons. They have no mass, so they travel at the same speed as well. They can never catch the car to push it. Like a game of tag where you run behind someone waving your arm at them but never quite able to touch them, you sure can’t push them harder if you can’t ever quite touch them. Nothing can catch up to provide more push.
So it can’t go faster than this “infinite” speed.
> How do we know that some immutable property of space/time causes light to travel at the speed it does rather than the other way around where some immutable property of light causes it to be incapable of traveling any faster
Because the speed isn’t how fast light goes, it’s how fast spacetime lets massless things go through it and light happens to be a massless thing. If mass is what resists movement, and you take all of it away, there isn’t any way to go faster because you can’t take more mass away than all of it.
(Wish I’d written this with whacking tennis racket on a car, bowling ball, baseball, ball bearing...) seems easier to imagine.