How does the non-degeneracy of the metric imply that gravity is not fundamental? In his General Relativity notes, on page 149, David Tong remarks that when we look for solutions to Einstein's equations, we can't just take any metric, such as $g_{\mu \nu} = 0$; we must pick one such that $\det g_{\mu \nu} < 0$ (with Minkowski signature). He writes further on this:

Other fields in the Standard Model don’t come with such restrictions. Instead, it is reminiscent of fluid mechanics where one has to insist that matter density obeys $\rho(x, t) > 0$. Ultimately, it seems likely that this restriction is telling us that the gravitational field is not fundamental and should
be replaced by something else in regimes where $\det g_{\mu \nu}$ is getting small.

This seems to be quite a deep and interesting observation, but I'm not sure I quite follow the logic. I can see how the requirement for positive matter density in a fluid comes from the fact that fluids are made of particles and we can keep on reducing, but we always need to impose a positive mass by hand to avoid coming up with unphysical solutions. Is Tong just saying that the fact that we need to add in a constraint by hand to match observation is a sign that there must be new physics?
 A: A little earlier in the same paragraph, Tong points out that there is a constraint on physical solutions of Einstein's equations
\begin{equation}
{\rm det} \left(g_{\mu\nu}\right) \leq 0
\end{equation}
In fact, the constraint is stronger, since you need one negative and three positive eigenvalues. (Except at a horizon).
This kind of "inequality constraint" is strange from the point of view of the Standard Model. There, you have constraints, but they are all holonomic (equations relating the fields and their derivatives).
One way to think about things is in terms of the path integral. In a gauge theory (like the Standard Model), you have a path integral that looks roughly like
\begin{equation}
Z \sim \int \mathcal D A \delta_\xi\left[G(A)\right] \Delta[G] e^{i S[A]}
\end{equation}
where $A$ is the gauge field, $S$ is the action, $\Delta[G]$ is the Fadeev-Popov determinant, and $\delta_\xi[G(A)]$ is a regularized delta function(al) enforcing the gauge fixing. In a unitary gauge, we fix the gauge completely such that all constraints are satisfied. Then, we only integrate over field configurations which satisfy the constraints (other gauges are equivalent because of the way we carefully set up the path integral).
In quantum gravity, you might think we need to do something like integrate over all possible spacetimes
\begin{equation}
Z \sim \int D g_{\mu\nu} (...) e^{i S[g]}
\end{equation}
where the $(...)$ is responsible for enforcing any constraints on the metric (modding out by diffeomorphisms, for example).
The point is that this inequality constraint $\det\left(g_{\mu\nu}\right)<0$ would require us to "cut off" the path integral in a hard way in unitary gauge, which is something doesn't happen in the Standard Model, because it doesn't have this kind of constraint.
With the major caveat that I don't really know what Tong meant, I believe the intuitive picture Tong has in mind is that the path integral "should" allow us to integrate over all metrics. Imagine a series of geometries in the path integral where the determinant of the metric changes sign. I think Tong is saying that rather than formulate our path integral so that the integral "stops" before the determinant changes sign, we should replace our theory of quantum gravity with something else, with more fundamental degrees of freedom, that is more like the Standard Model and doesn't have this kind of constraint.
While he doesn't say it, in string theory, in fact you don't formulate the path integral by integrating over the metric. There are different fundamental degrees of freedom (in some limits these are strings), and in principle using these should resolve singularities like the determinant of the metric changing sign (though we don't know how string theory does this, if it does). (In fact, there's not even an equivalent of the field theory path integral in string theory, since no one knows the correct non-perturbative definition, except in special cases).
Of course, this is all very speculative; no one really knows in detail how any of this should work in reality.
A: If we assume that Einstein’s theory is a principle theory and not a constructive theory then all solutions of Einstein's field equations should be physical, provided the corresponding boundary conditions are physical and correctly set. One cannot just pick up some solutions and throw out others without degrading the theory to the constructive one. The Einstein's field equations are second order differential equations on $g_{\mu \nu}$. The trivial solution $g_{\mu \nu}=0$ can be interpreted as the statement that without matter there is no spacetime and without spacetime no matter.
In my opinion, the physicality criteria should be imposed on theory's equations and corresponding boundary conditions and not on the solutions of the equations. The David Tong's formulation "we can't just take any metric" is in this context a little be misleading.
A: I had a brief look at Tong's notes. What he says is:

Let's start with $\Lambda = 0$. Here, the vacuum Einstein equations reduce to $R_{\alpha\beta}=0$. If we're looking for the simplest solution to this equation, it is tempting to suggest $g_{\alpha\beta}=0$. Needless to say, this isn't allowed! The tensor field, $g_{\alpha\beta}$ is a metric and as defined in section 3 must be non-degenerate. Indeed, the existence of the inverse $g^{\alpha\beta}$ was assumed in the derivation of Einstein's equations from the action.

This explains why he dismisses the so-called trivial metric, by definition it is not a metric. Moreover, were we to assume such a metric, then the topology would go haywire - it would be non-Hausdorff and not only have we lost the smooth structure, we have also lost the topological structure.
Now, the criteria for the non-degeneracy of the metric is simply that the determinant of the metric doesn't vanish. And because of the Minkowski signature and his choice of signs in the metric, this results in the constraint that this determinant is strictly negative.
It appears that Tong is suggesting that spacetime, rather than just being the theatre of where events happen, has an ontological weight of its own and should be treated as a substance in its own right. He is suggesting an analogy to fluid mechanics where a similar restriction holds and we can see this holds simply due to the continuity of fluids.
I don't see why it follows from this, as you also seem to think, that the metric can vanish somewhere. I would suggest that Tong is merely pointing out in the physics of today, where the possibility of new physics lie. Given that he is lecturing to graduate students of theoretical physics, some of which will be eager to explore the possibility of new physics, that does not seem inappropriate.
It appears he finds it implausible that if the metric can arbitrarily approach zero, then why cannot it be zero? Of course, as he points out above, this would mean that GR breaks down here. Thus he is mooting the possibility of new physics here. Then by continuity, we would also have this new physics where the metric is close to zero.
An analogy might help here. Consider Newtons inverse square law of gravity. If the distance between two masses is reduced to zero, then the gravitational attraction becomes infinitely large. This is unphysical. Hence, we  should expect new physics when two point particles approach each other - and we do: electromagnetic repulsion. That there was this repulsive force was deduced by the Croatian physicist, Roger Boscovich in the early 18th C.
