If you have a result stating that the non-linear completion of a free massless spin-2 field must be general covariant, then that means the action must only depend on curvature invariants like $R$, $R_{ab}R^{ab}$, $R_{abcd}R^{abcd}$, ...
Because we started with the action for a free massless spin-2 field, we can narrow down the allowed dependence on the infinitely many curvature invariants to be simply the Einstein-Hilbert term, $S \propto \int d^4 \sqrt{-g} R$. This is because the action for a free spin-2 field has only two derivatives, which means that only 1 power of the Riemann tensor $R_{abcd}$ can enter into the action. And the only invariant that contains one single power of the Riemann tensor is the Ricci scalar. No terms like $R_{abcd}R^{abcd}$ can show up because if they did, the action would be more than second derivative.
Certain combinations of curvature invariants can be shown to be simply boundary terms that do not affect the equations of motion, hence the ability to add Love-lock invariants.
The flip side of this is the linearisation of higher derivative curvature corrections leads to additional massive fields, i.e.
$S = \int d^4 x\sqrt{g} \left( a_1 R + a_2 R^2 + a_3 R_{ab}R^{ab} + a_4 R_{abcd} R^{abcd} \right)$,
can be shown to contain a massless spin-2 (the usual graviton), a massive spin-0, and a massive spin-2, http://link.springer.com/article/10.1007%2FBF00760427.
