Maybe time is more contiguous than continuous. Not moving, just re-appearing in a neighboring slice of Planck time like the procession of images on a 35mm film strip and the phenomena of after-images creating the illusion of continuous, smooth motion. Does that make sense?
It makes sense to me.
If I gave you a length standard and told you it was 1 meter long, but in fact it was 1.01 meters long, you could measure things forever without detecting that the standard was wrong.
Similarly, we are forced to measure durations using whatever we adopt as a measure of "time". It may just be that we can't observe any lack of continuity in time because of the tools we have to measure it.
The ancients believed that there were a few pure substances and that they consisted of "atoms" which, as the name implies, were indivisible. Using the tools they had, they were not going to be able to observe the particles inside atoms. Nor were they ever going to properly explain all the phenomenon that are the result of atoms not being indivisible.