General Relativity, as a metric theory of gravitation, works exactly like this. Einstein's resolution to the Hole Argument[0] was to realize that spacetime itself has no properties other than the intervals between events, or equivalently, the sum of every 4-distance between a state of matter and another state of matter.
In the standard cosmology, we use quantum fields that permeate the whole of spacetime and carry values at each point in spacetime. Some of the fields have values at each point that correspond to the presence of absence of a particle at that point (e.g. the electron field), while other fields provide a mechanism for one field to influence the state of another. In General Relativity we have the metric tensor, which is a field that carries information relevant to the causal structure of spacetime, and is determined by the values of the other fields (roughly, matter determines curvature, which in this case is a component of the distance in space and time from one configuration of matter to the next).
So, rather than saying spacetime doesn't exist, we say instead that spacetime is only relevant when there is a configuration of matter within it that can be used to established one or more systems of coordinates (e.g. the matter states converge, diverge or oscillate).
However, in Special Relativity there is no spacetime curvature (by definition) and so we do not need to use a metric tensor (we could naturally use the Minkowski tensor to describe the Minkowski spacetime, which is the spacetime of Special Relativity), but commonly attach a 4-vector to particles or bound collections of them. As with any spacetime in General Relativity, there is nothing special about a Minkowski spacetime that has nothing in it, or that has any other always-unchanging configuration of mass-energy.
The Standard Model of Particle Physics is a group theory with the Poincaré group as a subgroup; the Poincaré group is the isometry group of Minkowski spacetime, or in other words, in Special Relativity, when you move something following the rules of the Poincaré group, you do not change the fundamental states of the thing you are moving. The Poincaré group in 3+1 spacetime has 3 (bidirectional) linear spacelike translation components, 3 components of (bidirectional) rotation, 3 components of Lorentz boost, and one timelike translation. So if you do a particle-smashing experiment today and the same experiment tomorrow (or in a lab across campus), you will get the same result.
One can represent the action of the Poincaré group in several different ways. There is no harm in either approach in your second paragraph, as long as one is careful about how one manipulates either representation (especially, for example, swapping one representation for the other).
[0] https://www.wikiwand.com/en/Hole_argument
