No, the reason why we should seriously consider the interior of the Schwarzschild geometry is that all of the curvature invariants are finite at the event horizon, so it's obvious that geodesics can be extended beyond that point. The O-S solution is just one particular illustration of that general fact. No, we don't. A manifold is an open set. No open set extended into the Schwarzschild interior contains a singularity; the geometry is smooth everywhere. What indicates the presence of a "singularity" in the limit ##r \rightarrow 0## is that geodesics only reach a finite value of their affine parameters as that limit is approached. There is no discontinuity in the actual geometry; the limit point ##r = 0## is not part of the manifold. This is discussed in any textbook on global methods in GR, starting with the classic Hawking & Ellis and the papers that led up to it.