Space is not uniquely defined.
Spacetime, not space, is the physical domain in general relativity. Space is just a 3-dimensional surface within the 4-dimensional spacetime. The definition of a spatial surface (or "slice") is that it is a surface on which the time coordinate has a constant value. But there are infinitely many ways to pick a spatial surface and even more ways to pick a time sequence of spatial surfaces. None are objectively correct. So fundamentally we cannot make objective claims about what space is doing.
Positions within space are not uniquely defined.
The spatial slices were the surfaces of constant time. To complete our description, we also need to define the "threads" of constant spatial position. Here is an illustration of slicing and threading (taken from these lecture notes); $\eta$ refers to time, which is vertical in the diagram, while $x^i$ refers to position, which is horizontal in the diagram.
As with slicing, there are also infinitely many ways in which we can choose the threading.
Choosing coordinates in cosmology
The Universe is expanding. That means that things are moving apart in a uniform way. For many calculations, it turns out to be really convenient to use this expansion as the basis for our slicing and threading. In particular, we choose
- Spatial slices on which observers who follow the average motion of their surroundings (i.e., are "comoving with the Hubble flow") agree on how much time has elapsed since the beginning of the Universe.
- Time threads that follow these "comoving observers", so that they are defined to be at rest. Since the comoving observers are moving apart, these time threads diverge in the future direction.
Indeed, those coordinates are the only ones for which the cosmological principle (that the Universe is the same everywhere and in every direction) is satisfied.
Thus, we often choose to "let space expand" in the senses implied by these choices of slicing and threading -- particularly, the fact that the threads diverge. But that is a choice that we (and not the Universe) make. Other choices are equally valid. For example, when studying systems that do not expand, like galaxies and things inside them, it is much more natural to choose time threads that do not diverge. Then there would be no sense in which space expands.
Thus, expansion of space is not physical.
Since it is a choice that we (and not the Universe) make, it cannot have any physical consequences.
As an analogy, we like to describe locations on the Earth by their latitude (north-south position) and longitude (east-west position). Going north from the South Pole, the lines of constant longitude (meridians) diverge. Does this mean the surface of the Earth expands as you go north? There's no objective answer to that -- it depends on definitions. But if you were to walk north from the South Pole, would the diverging meridians induce (for example) a tendency for your body to stretch? Clearly no.