Right, I agree. There is one unique wave which occupies all of spacetime. I believe this is actually the standard view. For one thing, we know that reality is quantum, while this is a classical picture. I don't see what this concern of yours has to do with the above.
Light travels along certain paths, that's all that's needed.
We still agree, though I am about to take a detour. I'm glad we agree on this, because you seem to doubt it below. A worldline is just what we use for a field which happens to be localised infinitely precisely. In reality, there are no worldlines, because everything is really a field. I skipped a step.