> I am not really skilled in physics and have never understood why time would be a tangible dimension like, say, width or height, instead of a mathematical construction to argue about change.

The problem with your theory is that the main idea of Special Relativity is that time is a good tangible dimension, that has (almost) the same properties than the other three dimensions. And this is fully backed by experiments, and the effects are measurable in satellites, atomic clocks on planes, the color of gold, and many many many additional experiments.

One important property of the usual dimensions (x, y, z) is that you can mix them. Let's say that we choose z to be the vertical direction. Now we can choose x to be pointing to the east and y to the north. But we can mix x and y

x' = x * 1/ sqrt(2) - y * 1/ sqrt(2)
y' = x * 1/ sqrt(2) + y * 1/ sqrt(2)

Now x' and y' are obtained mixing x and y. You can think that someone else choose to point x in the north-east direction and y in the north-west. (I hope I got the signs correctly.) And all the experiments should be equivalent because the universe has no preferred direction, x and y are as good as x' and y'. [Since the Earth is spinning, we have a small technical problem here, but just stop the Earth to keep the discussion simple.]

Now, it is not a good idea to imagine that the east-west axis is a tangible dimension, but the north-south axis is something else. The main problem is that someone else can choose other directions, like (north-east)-(south-west) and (north-west)-(south-east) and get the same experimental results. Now, which one is the tangible dimension (north-east)-(south-west) or (north-west)-(south-east)? So we assume that all coordinates x, y and z are essentially the same thing and no one of them is special.

In most common situations, the vertical axis is different of the two horizontal axis. In normal situations, it is clear where it is up and down, so the "z" axis is special. It is not clear the direction of the "x" axis, and for most experiments you simply choose the direction that makes the calculations easier, but "x", "y" or any horizontal direction is as good as the other. [Just ignore again that the Earth is spinning and that it has a magnetic field. The spinning of the Earth and the magnetic field define two special directions that are both the "north" in some sense.]

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Now, even if there Earth is not spinning, and if the Earth had no magnetic field, the problem with the "z" direction going up is that two persons in different continents will disagree, about where is up and down. If you choose (x,y,z=up) in other continent will choose (x', y', z'=up') where x', y' and z' are obtained mixing x, y and z. And both will be totally convinced that "up" is special and there is a clear meaning of "up", in spite each one has a different "up". So it's better to assume that the universe has no preferred directions (x,y,z) and all the differences are due to the details, like a big chunk of dirt we call Earth.

If we assume that some of the directions x,y,z have some essential properties that the other doesn't have, it would not be possible to mix them and construct x', y' and z'.

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Now, about the problem with time ...

In Special Relativity you can mix the special coordinates and time. It's easier if you always multiply the time by c (the speed of light), and you use the coordinates (ct, x, y, z). Ad it's more simpler if you think that you measure the distance in x, y, z in light-years and that you measure the time intervals in t in years, so you get the same number in ct.

In Special relativity you can mix for example x with ct

ct' = x * ? + ct * ?
x' = x * ? + ct * ?

where ? means some coefficients like the 1/sqrt(2) in the x-y example at the beginning of this comment. It's easy to calculate them, but the exact numbers are not important.

The important part is that the formulas are mixing x and ct. In one mix you get ct' that is a new time-like coordinate and in the other mix you get x' that is a space-like coordinate. These are the coordinates that someone else sees when is moving at a different speed than you. You and the other person (in a train, plane, spaceship) will disagree about what how the time flows. You see t and the other person will see t' that is a mix of your t and your x. But both you and the other person can do any experiment and get equivalent results, because the universe don't prefer t to t' or vice versa. There is no experiment to determine if t or t' is better.

The main difference between the example with (x,y) and (ct,x) is that x and y are mixed in a slightly different way than x and ct. In particular, you can exchange x and y (or better x and -y for technical reasons). But you can't exchange completely x and ct. All the mixes have a new ct' that is somehow more similar to ct than to x, and a new x' that is somehow more similar to x than to ct. Moreover, all the observers agree that there is one time and three special dimensions, but they will not agree about how the time flows (as the will not agree where the directions x, y and z are pointing to).

The explanation of why can't mix "completely" x and ct is part of the mathematical details that I'm not writing here. It's somewhat related the fact that you can't move faster than the speed of light. Just get any introductory book about Special Relativity, but continue reading until you reach the chapter about Minkowski spaces. The initial formulas doesn't make too much sense until you reach Minkowski spaces.