The usual way to derive metric of a point mass in general relativity is (to my knowledge) based on assuming specific form of the metric that reflects spherical symmetry and independence on "time" (Schwarzschild metric).
I don't quite like this approach because the argument seems valid and natural, however after the solution is obtained, it turns out that it's incomplete. Complete metric is said to be given by Kruskal coordinates, usually they are introduced as a transformation of Schwarzschild coordinates.
I've found on the Internets a derivation of Kruskal metric directly from field equations. It's assuming spherical symmetry, but "radius" itself is not used as an independent variable and this accounts for it's implicit form and possible "time"-dependence. This approach in my opinion is much better, however it still leaves a feeling that something might not have been taken into account.
Is there a way to solve GR equations starting just with the fact that we have a point mass and that metric should be flat at infinity, without any prior assumptions about its form? Frankly, I'm not even sure how to write down stress-energy tensor without assuming something coordinates.