I'm not sure what the best tools are to treat the relativity of simultaneity. I find space-time diagrams (Minkowskii diagrams) convenient, but I think much of the value lies in learning to draw them, and less in looking at diagrams others have drawn. Looking at diagrams others have drawn can be helpful in learning how to draw them, of course. But, me doing the work to draw a space-time diagram for someone else doesn't seem like it will necessarily teach them anything. Also, it is a bit of a pain. And I'm a bit lazy, I suppose.

I suppose my thoughts on the relativity of simultaneity are that - I'd rather defver talking about it as long as possible, just because I know it confuses people. It is important, unfortunately, for explaining how it is possible for time dilation to occur due to relative motion without there being a state of "absolute rest". This seems like the main question of the OP, so perhaps it can't be avoided.

But using the principle of "avoid the relativity of simultaneity as long as one can", I have realized that I can provide some insight as to why time dilates for a moving observer, filling in the mathematical details of the rather lengthly scenario I outlined, putting in the numbers, and invoking the principle of invariants I espoused in my previous post, the Lorentz interval. This will accomplish the goal of showing why time moves slower for the moving observer, even if it won't necessarily quash the inconsistent-with-special-relativity idea that there is some sort of "absolute rest".

Rather than draw out a space-time diagram, I'm just going to list out some coordinates. It certainly might be easier to digest in a diagram form, but I'm going to encourage the reader who wants to see such a diagram to draw it themselves.

So to recap. We want to construct events occuring at regular time intervals for a stationary observer. With the Minkowskii construction, we can regard these events as lying on a worldline. Until we introduce the space dimension, we could just as well consider these events as lying on a timeline. Timelines are a common pedagogical tool for organizing events in contexts such as history and are not unique to physics. So, there shouldn't be any big mystery or glamour about just imagining these events as lying on a line. But it's convenient to give this line a name, we call it the worldline. And when we talk about moving observers, we'll need both time and space on one diagram. The standard diagram here is the Minkowskii diagram - I've read about a few other approaches, but I haven't used them.

Next, we want to construct the events on a moving worldline that occur "at the same time" as the events on the stationary worldline. We note that because of the relativity of simultaneity we've been talking about, "at the same time" depends on the frame of reference. But we won't stress the point much, we just want to make a note of it. The results we get will be valid and achieve our goal. We won't resolve here why there is no absolute rest, but hopefully even for someone who just can't accept that there is no such thing, we will show how utilizing the principles of invariants gives the correct result that the moving observer has lest time pass than the stationary observer.

Finally, we want to compute the ratio of the elapsed times between the events we've marked out for the moving observer and the same time interrvals for stationary observer. But how do we compute the elapsed time for the moving observer? We are aruging that time depends on motion - what is our guiding principle here to compute the time? Well, the guiding principle here is the one I mentioned earlier, the princple that the Lorentz interval is invariant for all observers. By computing the lorentz intervals, we can compute the time intervals.

Again, Taylor & Wheeler , "Space-time physics" covers this in a lot more depth than I will.

So: for the stationary observer, we get the following list of coordinates (t,x) where t is time and x is position (space).

$$(0,0) \quad (t,0) \quad (2t, 0) \quad (3t,0) ....$$

The list is easy - time advances in constant increments, and the position is always zero, at the origin.

And for the moving observer, we get a different list of coordinates (t,x)

$$(0,0) \quad (t,vt) \quad (2t,2vt) \quad (3t, 3vt)$$

The space coordinate is no longer constant - instead, because the observer is moving at some velocity v, the space coordiante is v *t.

To compute the Lorentz intervals, which will give us the proper times we are looking form, we need some simple formula. If we set the speed of light, c, equal to 1, by the proper choice of units, this is especially simple. For instance, we can use units of seconds as time, units of light seconds as distance, and measure all velocites as some fractional multiple of the speed of light. With this approach, we write:

$$ds^2 = dt^2 - dx^2$$

And we see that the Lorentz interval, and the elapsed time is the same. Thus we can conclude that if we compute the Lorentz interval for both the stationary and moving observers, we will have also computed the time interval.

If we don't make this simplification, we write instead

$$c^2 ds^2 = c^2 dt^2 - dx^2 \quad => \quad ds^2 = dt^2 - \frac{dx^2}{c^2}$$

It basically just makes things a bit more complicated. Is the extra complicity worth it for being able to use standard units? While I tend to think it isn't, I can see where someone might want to use standard units, so I'll include the calculations for that case as well. The result of this complication is that the Lorentz interval is no longer the same as proper time, instead, the Lorentz interval is a multiple of proper time. But, since our goal is to take the ratio of the moving observer to the stationary observer, this multiplying factor cancels out, and isn't of any significance.

The time interval for the stationary observer is obviously just equal to t from in the first list I gave. The time interval for the moving observer is ##\sqrt{t^2 - x^2} = \sqrt{t^2-v^2 t^2}## for the case of the moving observer, and the ratio we are looking for is just ##\sqrt{1-v^2}##, where v is expressed in factions of the speed of light.

If we go back to making c something other than 1, we can say that the Lorentz interval in the first case is c*t, and in the second case is ##\sqrt{c^2 - v^2t^2} = c t\sqrt{1 - v^2/c^2}##. So we get the ratio is ##\sqrt{1-v^2/c^2}## - basaically the same as before. In the first case, v was a dimensionless fraction of the speed of light, in the second case v has dimensions of velocity, so we need to divide it by c to get the correct dimensionless number.

This is just the usual relativistic formula. We can see that as v approaches the speed of light, the time between events is shorter for the moving observer than the stationary observer.

So to recap. While this simple calculation doesn't necessarilyi "put to bed" the incorrect idea of "absolute rest", it does illustrate that the principle of invariance of the Lorentz intervals makes the prediction that the time interval measured by a clock (proper time) between the events we marked out on a moving observers worldline is lower than the time interval masured byh a clock between the events we marked out on the stationary worldine.

To operationally be able to do special relativity, one simply needs to accept that the calculation works regardless of the state of motion, as opposed to only working in some "special frame".