“What exactly is spacetime? Is it a real thing like an atom, or just a mathematical construct that is used to describe how mass 'generates' gravity?” This is a false dichotomy. Atoms too are "just" a mathematical construct used to describe how certain observable, physical systems behave. It's even called the atomic _theory_, and like all other good theories it knows its limits. At low enough energies, it's too fine a…
The same applies to atoms, we all agree on what they are, and we can show pictures of them etc.
The question here is whether spacetime is something like that, or if it is a proxy concept with no grounding whatsoever in the universe, but with good predictive powers?
For example, for a long time we believed that Aether filled space, and although we now know this is not the case, the mathematical model provided by the Aether theory was pretty useful. You could very well imagine a world where we know Aether is not real, but continue using it for lack of a better theory as it is a useful framework.
> It's even called the atomic _theory_
Theory, in scientific theory, means the opposite of theory in our everyday language though. Theory does not mean a guess, it means a body of work accepted as valid and robustly tested.