Now, take an electrically charged object and measure the field at some distance. Then move it to one side, very quickly. How long will it take for the change in the field to register at your apparatus?
Indeed you did, but you *also* claimed that fields themselves don't propagate, something that is conclusively disproved by observing a field go from covering no area to covering a sizeable area.
What I did was to show an example wherein BOTH what you said about "changes" propagating as well as what I said about the field propagating were included in the same analogy.
And that analogy is also reproducable by experiment. We *can* observe that a magnetic field goes from covering no area to covering a sizeable area as soon as our electromagnet switches states from OFF to ON.
With that fact known, it is now reasonable to ask *how fast* did that magnetic field propagate from covering no area to covering its new, sizeable area.