I guess the answer depends on what approximations you are willing to make but we do not have a theory of quantum gravity readily available, so we can't really consider arbitrary quantum matter coupled with quantum gravity. (String theory is a sort of special case.)
If you are fine working with the classical limit of the standard model (which Andrzej Derdziński discusses in his terse book Geometry of the standard model of elementary particles), then you can couple it with gravity without a problem. It's just a few Yang-Mills fields. (Although, arguably with the classical limit, you may end up just investigating the EM field.)
But I am guessing the OP is interested in quantum matter content and quantum gravity. We don't really have a quantum theory of gravity.
Semiclassical considerations may be done, i.e., quantum matter content and classical gravity. This is investigated in Gordon McCabe's book The structure and interpretation of the standard model.
One extension of SM includes gravitons which are supposed to "mediate" gravity. As far as Susskind's lessons go, string theory is one possible way of including gravitons into being.
Well, the "graviton" can be understood as a perturbation of the metric tensor.
(This is how we generically think of particles in quantum field theory, as perturbations of a field.)
Deser proved (in the '60s?) it's okay, everything works out, and we using perturbations around even Minkowski spacetime works fine. (The intuition is basically the "Taylor series expansion about the Minkowski solution" converges for any perturbation.)
There's actually an intuition to understanding the terms from such a "Taylor expansion of the metric" as "Spacetime tells matter how to move, and matter tells spacetime how to curl up."
From the other end, even if we started with "an arbitrary spin-2 graviton" model, Feynman proved we end up recovering GR. (See Feynman's lectures on gravity for the sordid details.)
The moral is: there are many different ways to describe gravity, and at times some descriptions are more applicable to a problem than other descriptions.
How can then be a non flat metric in the first place if gravity could be brought in a coherent extension of the SM?
You can do quantum field theory in curved spacetime. Though the spacetime is "fixed".
For example, Stefan Hollands wrote a review paper computing the quantum Yang-Mills field in an arbitrary-but-fixed curved spacetime.
We can make it "slightly dynamic" by coupling gravity to matter "at the tree level". What this means is, when we draw Feynman diagrams of interactions, we do not permit graviton loops. This is "semiclassical gravity".
Another intuition for this is, we truncate the "Taylor series expansion" to the first term about some background (for terms coupling the metric to gravity). IIRC, the metric is still coupled to itself ("gravity still self-gravitates").
Thomas-Paul Hack's thesis reviews sordid details about the scalar and spinor fields in the semiclassical gravity setting.
But to turn on "full gravity" coupled to quantum matter, that is quantum gravity, which we don't have a theory of.