Supermassive black hole at the heart of the Milky Way is approaching the cosmic speed limit, dragging space-time along with it

That is correct only at non-relativistic speeds. You can read about special relativity on Wikipedia or something to get a better idea. To summarize: scientists tried to measure the speed of light. While at it, they measured the speed of light when moving really fast towards and away from the source. They found out it did not change no matter how fast you travel. So they rewrote coordinate transformations such that (condition 1) it kept the speed of light constant no matter the speed of your frame of reference and (condition 2) it approached the Newtonian (Galilean for the pedantic) coordinate transformation as your speed approaches zero.

When I say coordinate transformation you can think of an object moving at a speed of, for example, 100 m/s and an observer is moving at 50 m/s in the same direction. the observer sees the object moving at 100 - 50 = 50 m/s. If the observer moves in th4e opposite direction than relative to the observer, the object seems to move at 100 + 50 = 150 m/s. But according to the experiments, if something moves at the speed of light, no matter the speed of the observer, they will seem to move at the speed of light. so simple addition/subtraction will not work.

If you do the math and find a transformation that satisfies those 2 conditions you find the Lorentz transformation and it turns out if there is a constant universal speed then space and time must be relative. If you update your definitions of momentum and energy according to this observation you see that as you approach the speed of light, your kinetic energy approaches infinity. Usual Newtonian formula of kinetic energy is invalid at high speeds. I mean really really high speeds, near the speed of light.