I actually think the opposite is true. The way I've heard it phrased and explained that makes the most sense to me is "everything moves through spacetime at the same rate" - it's basically the clock speed of the universe. It's just that if you move faster in a space dimension that your relative movement in the time dimension slows down.
It only seems weird to us because our senses and minds evolved in an environment where things we can perceive never differ by relativistic speeds.
While I do like that intuitive explanation, it's lacking in describing all other aspects of the universe.
Like, how the energy required for an object with mass to approximate the speed of light in spacial dimensions goes to infinity, even though it's already traveling at that speed through spacetime.
Sure, one simple sentence is not going to explain the universe. But, at least from the simple relativity side of things, essentially everything falls out of (that is, it's a consequence of) that simple sentence. I.e. starting from that you can derive other consequences. E.g. "how the energy required for an object with mass to approximate the speed of light in spacial dimensions goes to infinity" is actually a direct consequence of that statement: every amount of energy you push into an object with mass causes it to accelerate, but due to the essential "clock speed of the universe", that acceleration is less and less as you approach the speed of light, and thus it takes an infinite amount of energy to reach the speed of light. Another way to think of it is that if it took anything less than an infinite amount of energy to reach the speed of light, then the speed of light couldn't be the universal speed limit, because you could add more energy that would accelerate it further.
On the other hand, my understanding is that quantum mechanics is another beast entirely, and one of the biggest problems in physics, and to developing a "theory of everything", is to unify quantum mechanics with general relativity.
Although nothing will explain everything, still it's fine with the first point: increasing the rotation vector of momentum in spacetime increases mass. The rest follows, since you know that the more mass, the more energy required to accelerate still more.
But if you are interested, a significant amount of the basics of quantum mechanics follow directly from Fourier transforms -- which unfortunately are harder to self-study than spacetime rotations.
I came up with a variant of this that extends the motion vector into “matter” dimensions. Then the logical consequence is that the more matter you have, the less of the unit vector is available for movement in time == time dilation due to mass.
Similarly, only vectors with zero length in the matter direction can have unit length in the space/time direction. This is the “only massless particles move at the speed of light” rule.
FWIW there are other answers in that Stack Exchange that, in my opinion, give a better description of the situation, and in my opinion the primary objection in that particular answer is the definition of "move". Fair enough, but I think it's still a helpful description for laypeople who are not fully versed in the math.
It only seems weird to us because our senses and minds evolved in an environment where things we can perceive never differ by relativistic speeds.