I obviously cannot answer for the descriptions that other people give models and analogies in their explanations, but they should be considered with care and accepted with reservations in general. All descriptions and illustrations are essentially analogies, and not to be confused with the object described. Ultimately this remains true even if I hold up an egg and say: "This is the shape of an egg". After all, all you perceive is the light or sound reflected from the egg, or the indentations that the egg makes on your skin when you feel it.

Even if you argue that a mathematical description would be the true representation, you would be wrong for a number of reasons that I hope you will find to be obvious on reflection.

In your example of representing gravity in a distorted two-dimensional surface, remember what the point of the representation is. It is as much an analogy as a two-dimensional perspective drawing of a tesseract is in representing the representation of a four dimensional cube as a shadow or projection in three dimensions. The distortion in the representation of the surface has nothing to do with representing gravity as gravity. For one thing, you will notice that the direction of the gravity, namely towards the centre of mass of the object in question, is not perpendicular to the distorted surface, whereas the direction of distortion is.

You suggest that the models present some poor comparisons. Now, why do you suppose that creators of such models should have been so slapdash in their creation of comparisons? Surely it would be a lot easier for the silly people who came up with that silly representation simply to represent gravitation and time directly, thereby avoiding unnecessary self reference?

It seems to me that the following suggestion is long overdue: As soon as some people in this forum can present us with such pictures (or even three-dimensional random dot stereograms) I am sure that the rest of us would be delighted to adjudicate the best effort. Just remember that self reference will be grounds for disqualification. No prizes are offered for anything so trivial, but a good deed is its own reward, and I am sure that our praise will be appreciated!

Your asking the question of the  century.

There is no explanation of gravity.

The self referencing weight of the ball, distorting a frame of time, as in the illustration, doesn't help much. As you point out, it presumes the force of gravity in trying to illustrate it.

It wasn't necessary to diagram spacetime to navigate to the moon and back. They did it with a computer that was less powerful than todays pocket scientific calculator.

All we've got is the inverse square law, and that's just, how to deal with it, not an explanation of why the force exists.

There's no explanation of why magnetism and electricity exist, we describe them, we measure them, we make them serve us, but really have no "explanation".

As a teenager, I thought we had it all figured out when they described the atom, the protons, neutrons, electrons, and built the atom bomb. Now we name sub-atomic particles and search for the God particle, when all we can do is peel off another layer of the onion as we try to look into the "mind of God'.

It seems, God is a word we use when we reach the limit of our understanding.

Illustration - wikipedia - spacetime