Is the assumption that space-time has to be a continuum just a matter of mathematical taste? Isn't there any physical significance associated with it?
Underlying this question is an erroneous assumption about what physics is and what it isn't.
The point is that physics is a description of reality. However, it does not say anything about whether the description equals reality.
In particular, nobody knows whether space-time truly is continuous or not. But continuity sure describes it extremely well. Also note that this description might change in the future when we can do even more precise measurements.
Space and time are inclusive notions. Many do not know it but it is so.
When you tend the number of information bits (measurement points, pixels, if you like) to infinity, then certainty emerges. Space and time are those certainties.
As inclusive entities, they are not fundamental, primary but secondary. In other words, they are illusions. Elementary, fundamental are those bits of information which constitute the whole picture.