In the classical limit, the laws of physics should be causal and deterministic, i.e., initial and boundary conditions should uniquely determine the behavior over spacetime, up to diffeomorphisms (and gauge transformations). This is generally achieved by writing a classical field theory in terms of a principle of stationary action.
In the Einstein field equation $G^{\mu\nu} = 8\pi T^{\mu\nu}$, the left-hand side comes from variation of the gravitational action and the right-hand side comes from variation of the matter action -- each variation being with respect to the metric $g_{\mu\nu}$. For a deterministic solution, we must have some specific form of the stress-energy tensor $T^{\mu\nu}$.
If all we specify about $T^{\mu\nu}$ is that it obeys the local conservation law $\nabla_\mu T^{\mu\nu} = 0$ and is otherwise unrestricted, then this doesn't tell us anything about the geometry of spacetime, because $\nabla_\mu G^{\mu\nu} = 0$ is a mathematical identity. To have a deterministic theory, we should explicitly derive $T^{\mu\nu}$ from a matter action, and should also include the matter equations of motion from varying the matter action with respect to the underlying matter fields.
A rough argument that such a theory achieves determinism is as follows: Each matter field has a corresponding equation of motion from varying the matter action with respect to that field. Meanwhile, there are 10 independent components of the metric $g_{\mu\nu}$, whereas $G^{\mu\nu} = 8\pi T^{\mu\nu}$ provides only 6 independent equations for these components (because applying $\nabla_\mu$ to both sides shows that 4 of the nominal 10 components of the Einstein field equation are identities). The overall result is that we have 4 fewer equations than unknowns, which is exactly right to allow for an arbitrary diffeomorphism with 4 components that does not affect the physically observable behavior.
Thus, I disagree with the statement that
GR in its purest form allows spacetime curvature to exist without some external causal agent
since a very important property of GR is its determinism in the presence of a well-defined matter action (including the absence of matter, giving $G^{\mu\nu} = 0$, which is arguably actually "GR in its purest form").
Note 1: Technically, some things can be said about the geometry of spacetime under inequality constraints on $T^{\mu\nu}$ known as "energy conditions", which are fairly general. The resulting conclusions, though, are about overall structural features (such as singularities) and do not by themselves provide a deterministic solution.
Note 2: A degenerate case where we can write a "matter action" without any matter fields is the cosmological constant $\Lambda$: The matter action is proportional to the spacetime volume, and the stress-energy tensor is $T^{\mu\nu} = \Lambda g^{\mu\nu}$. This automatically satisfies $\nabla_\mu T^{\mu\nu} = 0$ without requiring any matter equations of motion. It can be also be considered an addition to the gravitational (as opposed to matter) action.
Best Answer
It is completely possible for DM to have only gravitational interactions, and this is "nightmare scenario" is well-known by practitioners.
There are at least three hints against this possibility. First, there are a lot of astrophysical and cosmological anomalies (such as the galactic center excess) that could each be explained via some non-gravitational DM interaction. Second, for reasons of economy, we typically prefer theories of DM that can also explain outstanding problems in particle physics (such as the strong CP problem). Third, a theory of DM generally needs a way to explain how the DM was produced in the early universe, and most production mechanisms we know about (such as freeze-out) need non-gravitational interactions. However, it could still turn out that all those astrophysical anomalies have other explanations, that DM is completely unrelated to all problems in particle physics, and that DM was produced purely gravitationally.
If DM takes the form of black holes, they could interact only gravitationally and yet still be directly detectable, either through their Hawking radiation (for lighter ones) or through gravitational lensing (for heavier ones). But if DM takes the form of particles interacting only gravitationally, we're out of luck; we probably could never detect such particles directly.
In that case, the only way to test the DM hypothesis would be to do more precise measurements of its gravitational influence on galaxies and cosmology. This would be perfectly legitimate science, as there are plenty of objects in science that we've never observed "directly" but which still have great explanatory power. For example, astronomers living today have never and probably will never go to another star, but the hypothesis of stars is still universally accepted because they do a great job of explaining the points of light we see in the sky. Still, if DM ends up like that, it would be quite frustrating and disappointing, because in most theories DM is already right here, regularly passing through our planet.
In this case, particle theory would probably not say very much, because the models would be very unconstrained. It's just not that hard to add in extra particles that don't interact with the rest. If this happened, particle physicists would probably just step away to work on other problems.