Let me first go back to the theory of electromagnetism.
I will first discuss the context of electromagnetism, and then, using that as starting point I will move to the context of general relativity.
Maxwell's theory has been superseded by quantum theory, but as we know: Quantum theory should be thought of as a superset of Maxwell theory. Maxwell theory has its domain of validity, and within that domain of validity quantum theory confirms the validity of Maxwell theory.
In terms of Maxwell theory all of space is filled with an entity, a medium, that in the absence of electrostatic charge is in a uniform state.
Maxwell thought of this medium as the luminiferous aether. In terms of relativistic physics that Maxwellian concept is replaced with a concept such that Lorentz invariance is inherent to it. Obviously the introduction of inherent Lorentz invariance was new, but other than that: all properties that Maxwell theory attributes to the medium were ported to the superseding theory.
In the absence of electrostatic charge the medium is thought of as in a uniform state. In this uniform state the medium does not affect motion of particles. The presence of electrostatic charge sets up a state of stress in the medium, and this state of stress has a physical effect: a charged particle interacts with that stressed state, and that affects the state of motion of that charged particle. As a consequence charged particles will start moving towards each other (or away from each other, depending on the sign of the charge.) We call that state of stress 'the Coulomb field', or 'the Electrostatic field'.
I use the expression 'stress in the medium', that's just a metaphor. The idea is that this stressed state is a state of tension as compared to the uniform state, and that it is inherent in the medium that once the charged particle recedes the medium will revert to the uniform state.
The medium has another crucial property: the change of state (stressed/uniform) is itself a form of motion, and this motion has the property that once it gets going it tends to keep going.
Oscillation has two requirements:
a tendency to revert to some lowest state
a tendency of existing motion to keep going
As we know: the mediator of the electromagnetic interaction meets the above two requirements. As a consequence the electromagnetic field is capable of propagation of waves in the medium itself. As we know: we call these propagating oscillations of the medium: electromagnetic waves.
Maxwell derived the speed of propagation of electromagnetic waves from first principles, and he found it to be so close to the known speed of light that he concluded that light must be propagating electromagnetic waves.
Now to the question: is this electromagnetic medium real, or a mathematical convenience?
I don't think a definitive answer to that is possible.
The aspect that is actually observable is motion. A visual example of that is the images of particle tracks in cloud chambers. If the particle track is curvilinear it must be charged particle, deflected by the magnetic field that is applied. There are ways to infer the velocity, etc. etc. All the detectors of the particle accelerators: the data that are collected is state of motion of particles.
The inferred field cannot be observed directly. What is observed is how the inferred field affects motion of particles.
Maxwell theory and quantum mechanics describe interactions between particles as mediated by a field. But we have no way of observing this mediator directly, the existence of the mediator is inferred.
In the case of electromagnetism we have that electromagnetic waves continue to propagate even after the original source has ceased to exist. That is strong corroborating evidence that the mediator of electromagnetic interaction is physically real.
The spacetime of General Relavity
In terms of general relativity we have as starting point that is is assumed that there is a mediator of gravitational interaction, but in the absence of a source of gravity this mediator is in a uniform state. In this uniform state the medium does not affect existing motion of particles; they continue in a straight line. However, this uniform state has the following property: if another interaction, for example electrostatic interaction, is active, trying to make a particle deviate from motion in a straight line, then there is opposition to that. the usual name for that property of spacetime is 'inertia', of course. The opposition to acceleration with respect to the local spacetime is proportional to the acceleration.
We describe this uniform state of the mediator of gravitational interaction as 'geometrically flat spacetime'. We call it: Minkowski spacetime.
The presence of gravitational source sets up a biased state in the mediator, and in this biased state there is a physical effect. The effect is that particles acquire a velocity towards the gravitational source. This biased state of the mediator is referred to as 'curvature of spacetime'. When a gravitational source recedes the mediator reverts to uniform state.
Now to a specific case: a binary system of two celestial bodies. The two members of that binary system are moving through spacetime. As a celestial body moves local spacetime is put in a state of stress; as the celestial body recedes again the spacetime reverts to as close as possible to uniform state.
According to General relavity:
Spacetime being put in a state of curvature, and reverting towards uniform state, is a form of motion. This motion has the property that once it is going it tends to keep going.
Thus the mediator of gravitational interaction meets the requirements to be capable of propagating waves in the medium itself.
According to General Relavity a binary system will lose orbital kinetic energy because it is generating gravitational waves that propagate out into the Universe.
By the 1980's it was confirmed that the decay in the orbital period of the Hulse-Taylor binary system is consistent with the amount of loss of kinetic energy predicted by General relativity.
Gravitational waves continue to propagate after the original source has ceased to exist. The LIGO experiment has been succesful multiple times in recording gravitational waves from the merger of a binary system of two black holes.
After the merging event the system no longer emitted gravitational waves. A couple of billions of years later the LIGO detectors registered a propagating undulation of spacetime.
Just as with any field theory: the existence of the field is inferred. What is observed is motion. What is observed is how the inferred field affects the motion of particles
The key points:
General Relativity is a field theory; General Relativity does assert that there is a mediator of gravitational interaction.
The hallmark of a field theory is that there is a supposition of a mediator of the interaction that responds to the presence of a source.
In the case of gravitational interaction the nature of the mediator is unique to it.