Sometimes, we call "misconceptions" the opinions others have, that prevent them to understand our ideas, or to agree with us. These are not necessarily wrong conceptions, but they tell a different story, which is not compatible with our story.
Quantum mechanics works perfectly fine with continuous space and time, and each point can be identified by real numbers. It works fine by evolving a state vector in time, and everything we know about quantum phenomena, was predicted by thinking about that state vector from a Hilbert vector space (over a continuous field), and that this vector evolves in a continuous time, driven by a unitary evolution operator.
But, the author of the paper wants indeed to say something. The point is that we don't know about the continuum of positions and moments from experience. According to the author of the paper, experiments can only inform us about a finite number of events, which taker place in a finite number of positions, at a finite number of moments in time. At least, this is what I understand from this quote:
the spatial properties of things and the times at which they are
possessed exist to the extent that they can be inferred from facts,
and facts do not warrant the inference of an infinitely differentiated
space or time. Between the factually warranted times that make up the
history of a system, there are gaps; there are “periods of time” that
do not exist for the system (Sec. VI). Everything that can be said
about those periods is counterfactual.
In my opinion, this is a healthy skepticism. We should doubt anything we can't experience*.
*On the other hand, we can't deny the success of the standard quantum mechanics, based on continuous space and time. Also, theory of relativity supports the continuity of spacetime, and breaking this continuity, means breaking the Lorentz invariance. But I just wanted to explain author's claim, not to enter in a debate.