That equation describes a curve in 2 dimensional space with no differences between points with each point having particular time and space coordinates. And you can even resolve it to express time as a function of space. But we do not interpret that as a flow of space.
Sure, but you still can't explain it without adding an extra dimension to your model, and that extra dimension is not symmetrical with the space dimensions. In fact, there is no physical model that I am aware of that makes time and any space dimension interchangeable.
The fact that our mathematical models generally seem to allow movement backwards and forwards in time is in stark opposition to observed physics. This obviously suggests that the models are wrong, and there is something about time we are not capturing well enough. Relativity does somewhat fix this, by essentially postulating that everything is moving with speed c in space-time, but that is a somewhat unsatisfying explanation, since it seems natural to ask why.
The extra dimension does not explain the movement or change. The world line corresponding to the equation is static. It does not show at all why when one looks at a particle, one perceive the movement and change in the position, and not, for example, the whole world line at once.
I am not saying that time and space are the same in physical equations. The point is that they do not describe the perception of time flow, not that there is no difference between time and space.
As I wrote the structure of equations (equations are parabolic or hyperbolic, not elliptical, the sign of the time dimension in the metric tensor is opposite to the space coordinates) reflects that one can predict across time but not space. I.e. the equations reflects that from a picture of a room one can tell what will happen in one hour or what did happen one hour ago. Shadows from the Sun will move, a sleeping cat will not be there, but things will be mostly the same. But try to tell what is in the rest of the room from a one-hour long video of the wall. It is not possible.
But this difference tells nothing about perception of “now” or the time flow.
In a sense the equations reflects how memory operates. We remember a sequence of events and we can focus on a particular moment or select events in an arbitrary order similar how we can select a point or points on the world line of a particle. But the memory does not have “now” and so the equations reflecting the notion of the world line tell nothing about the time flow.
