Well, first of all you should ask yourself what do you expect from the concept of energy and momentum. Or in other words, what is energy and momentum, really? You have a set of intuitions in mind, but the minimal requirement is that these are quantities that are 1) conserved, and 2) reduce to our "usual" definitions of energy and momentum in suitable limits.
Let us take a look at the Einstein equations
$$G_{\mu\nu} = 8\pi T_{\mu\nu}$$
The Einstein tensor $G_{\mu\nu} = R_{\mu\nu} - R g_{\mu\nu}/2$ does have the property of being conserved $G^{\mu\nu}_{\;\;\;;\nu} = 0$, same as the stress-energy tensor. Furthermore, and this is a somewhat nuanced point, certain parts of it can be interpreted as gravitational energy in the weak-field limit.
For instance, in the post-Minkowski limit you have $g_{\mu\nu} = \eta_{\mu\nu} + \epsilon h^{(1)}_{\mu\nu} + \epsilon^2 h^{(2)}_{\mu\nu}+...$. At first order in $\epsilon$ the left-hand side of the Einstein equations just correspond to a linear operator acting on $h^{(1)}_{\mu\nu}$. At second order, however, you have the same operator acting on $h^{(2)}_{\mu\nu}$ and terms quadratic in $h^{(1)}_{\mu\nu}$ that can be interpreted as (minus) the stress-energy tensor of the field $h^{(1)}_{\mu\nu}$ (sourcing the gravitational field correction $h^{(2)}_{\mu\nu}$). So $G_{\mu\nu}$ (or $-G_{\mu\nu}/8\pi$) seems to naturally offer itself as a gravitational stress-energy tensor.
However, as you may have already noticed, when $-G_{\mu\nu}/8\pi$ is moved to the right-hand side of the Einstein equations as a part of stress-energy, then the interpretation of the equations ends up being that the total stress-energy content of any point of space-time is exactly zero. Every order of the post-Minkowski expansion is really just enforcing that statement, at higher and higher order in $\epsilon$.
This may be elegant, but underwhelming, especially in vacuum space-times. In vacuum space-times, such as space-times of inspiraling and merging black holes, a lot can be obviously going on, while this interpretation is just telling us there is no stress-energy ever flowing, present, or exchanged. This is the reason why other notions of energy and momentum of the gravitational field have been introduced. But I am going to be honest, their real value is in keeping track of how the gravitating mass (and/or momentum) of a system evolve in time. After this violent process with outgoing gravitational radiation, how much will this system still attract me gravitationaly? These "nonlocal" formulas mentioned in the other posts here provide the answer. But beyond this pragmatic meaning, I would defer from interpreting them too deeply.
