General Relativity is a mathematical framework within which we can construct Lorentzian manifold models of reality. In general, structures (e.g. the spacetime manifold) of a given model taken to represent some aspect of observable reality need not be physically real aspects of nature, whatever that might mean. Indeed, they're almost certainly not-- what are the odds that the formalism we decided was most natural to us perfectly captures the nature of reality? In practice, all we can ask of a model is that it provides a means of unequivocally (within discernible error) predicting some observations. We like (accurate) models that are broad in scope, offering a means of predicting many different types of observations, and that sit well philosophically within the larger network of other successful models, but even these cannot be presumed to present "actual" reality.
All that to say: the only crucially significant feature of a model is the collection of predictions it makes. Details of a model's structure that don't impact its predictions are ultimately of little import, especially if they also don't impact the model's philosophical interpretation. The particular set of cardinality $2^{\aleph_0}$ one chooses to think of as underlying the Lorentzian manifold of a GR model is probably one of the least impactful (both observationally and philosophically) features I can imagine, and for this reason, GR makes no such choice outright-- any one at all will do.
A comment worth making is that the set in question is essentially universally given an interpretation as the set of spacetime events, pairs of "space" and "time" instances that characterize where and when something can occur, but this is only a heuristic intuition, not a rigorous definition that nails down the set theoretic object under consideration. Since all candidate sets are bijective, such an interpretation on one of them induces the same on all others.