To be precise, I would make the above changes in bold.
It's not that Time Dilation is not real-it's that it is frame dependent which means that we cannot establish an absolute reality to it. Clocks can change their actual rates when they accelerate, it's just that we cannot say that a clock is Time Dilated after it accelerates and not before or vice versa unless we specify an IRF.
Let me illustrate with a couple spacetime diagrams based on a modification to the ones in post #10. Let's say in the first diagram that the clock started out traveling to the left at 0.6c and that at a time of 4 seconds, the clock changes to being at rest:
Now we will transform this scenario just like we did before and we get:
Now looking at these two diagrams which represent exactly the same scenario, we see that the clock is Time Dilated to begin with in the first diagram and ends up being Time Dilated in the second diagram.
We can even find an IRF in which there is no change in the clock's Time Dilation as a result of the same acceleration:
This IRF is moving at -0.333c with respect to the first one which results in the clock changing direction but maintaining the same speed before and after the acceleration.
There are even IRF's where the clock is Time Dilated before and after the acceleration to different degrees. We can never say what the Time Dilation of a clock is without specifying the IRF.
Because the muons are at high speed in the Earth frame and at rest in the lab. In the rest frame of the muons entering the Earth's atmosphere, they are not Time Dilated while the ones in the lab are.