They nearly have it in the article but don't take the next step, which is to realise that time is gravity. We are falling towards the future at the speed of causality. You can slow your experience of that by travelling really fast, or by being near something with a large gravity. Quantum particles can simply ignore gravity and time while we are forced to feel their effects due to our size, like a mote of dust can ignore gravity, but a brick can't.
Well, almost. Because these usual descriptions you give here are approximations. Yes, to the first order, gravity is expressed as the effect on time in general relativity. So you could describe how planets move simply by calculating how they move differently through time. But in the full picture, the way you move through space or near strong gravitational fields also influences your experience of space itself. So the ultimate realization is that gravity is space and time. Or spacetime. Basically exactly what the article says when it references general relativity. And it holds as well for quantum particles. They experience spacetime just as well as we do. The ultimate question is, does spacetime itself also come quantized when you look close enough? This is the true head banging one that noone knows.
A mote of dust doesn't ignore gravity in a vacuum ;)
You might be interested in The Order of Time by Carlo Ravelli. Time and gravity are certainly linked, but from what I took away from the book (which is a lot to digest, even as non-mathy as it tries to be) is that Time is really heat. Heat moves only from hot to cold, dispersing in some entropic fashion as we move toward the final state of the universe, but in the meantime we can measure that time/heat "flows" at different rates, depending on how near or far you are from large bodies.
I likely need to reread it, though, as some of its ideas are a bit above my weight class when it comes to understanding physics. But, you may enjoy it!