Does gravity make time flow?

I'm not a physicist, but I do like physics. There are a lot of nice ideas in physics that can be beautifully modeled by awesome mathematics. This I think is the reason why most physics people think about math as a tool, rather I have the math person  point of view that math works for every world, specially for this one.

I don't want to start the third world war in between physicists and mathematicians (which is more like a cold war at the moment), but I want to express some ideas based on my mathematical imagination, that is, I have no physical intuition or base to state these.

It all started almost a year ago when, for some weird reason, I started spending a lot of time thinking about: time. It had always fascinated me this concept for which we don't really have a definition, we just take it as a parameter. We know that it runs, that it goes forward and that we cannot rewind. It was with Einstein that we took a step forward into understanding this mysterious being. We now know that time runs at different speeds depending on your reference point, but still we have that time can only run in one direction, that is, we cannot reverse it. Again, I am no physicist, and my little knowledge is that you can explain this also from the point of view of thermodynamics, but I was trying to get my own explanation, my own SciFi reason of this.

Basically my question was What makes the time to run?, Why there is no similar effect on the spacial coordinates? My approach to these was in a more geometrical way. Supposing that we start with a 4 dimensional real space
$$M=\{(a,b,c,d)|a,b,c,d\in\mathbb{R}\}$$,
we have that one of the variables serves as a parameter for the other 3. This is the mathematical way of saying that one of them is time. In other words, what is happening to get a time variable is that we are parametrizing a curve in $M$.

When talking about curves or paths jointing two points in space, it is almost immediate to think about geodesics. These arise when there is some sort of minimization process, and nature really likes this kind of property.

Following this train of thought, the responsible for creating geodesics in $M$ has to be a metric $g$ on $M$. Then geodesics are the result of minimizing the arc length of paths joining two points in space taking into account the action of the metric $g$.

Since the metric $g$ is the mathematical connection to incorporate gravity in $M$, ultimately time is a consequence of gravity.

That is, gravity provides a way to minimize paths in $M$, but still we need two points in space to make sense of this model.