The dimensionally reduced Einstein-Hilbert action of a spherically invariant metric is a special case of two-dimensional dilaton gravity. This type of theories was the subject of a quite extensive research at least in the past decade or so. It has applications in integrable systems, quantization of 1+1 gravity and it (what is relevant to this question) offers an alternative proof of the Birkhoff's theorem (: The Schwarzchild solution is the unique spherically symmetric solution of the vacuum Einstein equations). In many of these works the Schwarzchild solution was obtained as a byproduct as the solution of the 2-d field equations
Technically it is not very easy since it requires the solution of nonlinear partial differential field equations in 2D. In the following reference by Marco
Cavaglià the derivation is quite transparent, which can be summarized as follows:
Starting from the general spherically symmetric 3+1 metric:
$$ds^2 = \phi^{-\frac{1}{2}}g_{\mu\nu} dx^{\mu}dx^{\nu}+\phi d\Omega^2$$
(the indices run over a Lorentzian 2-space). After the substitution in the Einstein-Hilbert action and the integration of the angular coordinates, the following reduced 2d theory is obtained:
$S = \int d^2x \sqrt{-g}[\phi R^{(2)}(g) + V(\phi)]$, with ,
$$V(\phi)=\frac{1}{2\sqrt\phi}$$
The field equations:
$$ (\nabla_{(\mu}\nabla_{\nu)}- g_{\mu\nu}\nabla_{\sigma}\nabla^{\sigma})\phi + \frac{1}{2}g_{\mu\nu}V(\phi)=0$$
$$R+\frac{dV}{d\phi}=0$$
The nonlinear field equations can be solved by means of a Bäcklund transformation:
$$ \nabla_{\mu} \psi = \frac{\nabla_{\mu}\phi }{\nabla_{\sigma}\phi\nabla^{\sigma}\phi}$$
Leading to the following linear field equations
$$\nabla_{\sigma}\nabla^{\sigma}\psi = 0$$
and
$$\nabla_{\sigma} M = 0$$
with
$$M = N - \nabla_{\sigma}\phi\nabla^{\sigma}\phi$$
and
$$N = \int^{\phi}V(\phi\prime)d\phi\prime$$
The linear field equations can be solved in the conformal gauge:
$$ds^2 = \rho(u,v)dudv$$
as:
$$\psi = U(u) + V(v),$$ and $M=const.$
The solution for the metric is obtained from the Bäcklund transformation equation:
$$\rho^2 = \partial_u\phi \partial_v\phi\partial_u\psi \partial_v\psi ,$$ or
$$\rho= (N(\phi)-M)\partial_u\psi \partial_v\psi ,$$
and
$$\frac{d\psi}{d\phi} = \frac{1}{N-M}$$
By performing a coordinate transformation to the independent variables $U$ and $V$ instead of $u$ and $v$ respectively. We obtain:
$$\rho= N(\phi(U+V))-M)$$
Again changing the coordinates to T=U-V and $\phi(U+V)$, we obtain:
$$ds^2 = (N-M)dT^2+(N-M)^{-1}d\phi^2$$
which is the Schwarzchild solution for $N=\sqrt\phi$.
Please observe that the metric was not assumed to be stationary, thus this derivation consists of a proof of Birkhoff's theorem.
